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Two Equally Valid Interpretations of the Linear Multicompartment Mammillary Pharmacokinetic Model
Two Equally Valid Interpretations of the Linear Multicompartment Mammillary Pharmacokinetic Model JAMES R. JACOBS*", STEVEN L. sHAFER*, JANINE L. LARSENI,
AND ERICD. HAWKINS' Received May 22, 1989, from the 'Department of Anesthesiology, Duke University Medical Center, Durham, NC 27770, the *Departmentof Anesfhesiol y (7 12A), Stanford University, Veterans Administration Medical Center, 3801 Miranda Avenue, Palo Alto, CA 94304, the SPritzker Institute of %dim1 Engineering, lllinois Institute of Technology, Chicago, IL 60616, and the 'Department of Pharmacology, East Carolina University, School of Medicine, Greenville, NC 27858. Accepted for publication August 8, 1989. __ _ _ _ _ ~ Abstract 0 In pharmacokinetic modeling it is common to use compartmental structures to describe the disposition of a drug in the blood or ~
~
plasma. Typically, a linear multicompartment mammillary model is equated with the multiexponential equation derived from observing the decay of the plasma drug concentration following an intravascular injection.Classically,the mammillary models are constructed so that the concentrationsin each of the compartments areequal at steady state,the apparent volume of distribution at steady state is equal to the sum of the individual Compartment volumes, and the apparent volume of each peripheral compartment is equal to the ratio of its intercompartmental rate constants times the central compartment volume. On the basis of what can be measured in the plasma, however, it is equally valid to assume that the sizes of the peripheral compartment volumes are equal to the central compartment volume and that the steady-state concentration in each peripheral compartment is equal to the ratio of its intercompartmentalrate constants times the Concentration in the central compartment. In fact, these are but two of an infinite number of interpretations of the peripheral Compartment volumes. __ -. .-
A
B
~
Multicompartment models are frequently used to provide an abstraction of the pharmacokinetic behavior of drugs whose disposition in plasma following a rapid intravascular injection can be described by a multiexponential equation. The implications of the compartmental structure are much more intuitively appealing than those of a multiexponential equation. If drug concentration can be measured only in the central compartment, then the volumes of, and the drug concentration in, the peripheral compartments cannot be determined; but, it is generally assumed that the concentration in each of the compartments is equal at steady state. Mathematically, this implies that the steady-state volume of distribution is equal to the sum of the individual compartment volumes. We will show that on the basis of what is measurable (i.e., the concentration of drug in the blood or plasma as a function of dosage and time) it is equally valid to assume that the compartment volumes are equal and that, therefore, the apparent concentration of drug in the compartments a t steady state is proportional to the ratio of the intercompartmental rate constants.
Theoretical Section
Figure 1-(A) Linear two-compartment mammillary pharmacokinetic model; ( 6 )linear n-compartment mammillary pharmacokinetic model.
C,(t) = Ale-A1L + A2e-A2t
(1)
where A, and A , are in units of concentration and A, and A, are in units of reciprocal time. The pharmacokinetics of the drug is assumed to be linear and, on the basis of linear system theory,' eq 1(scaled by D-')represents the impulse response of the system, relating the input (drug dosage) to the output (concentration of drug in the plasma). The biexponential equation has a one-to-one correspondence to a linear two-compartment mammillary model (Figure 1A). where drug administration. measurement. and elimination occur exclusively through the central compartment. The parameters of the two-compartment model are related to the coefficients of the biexponential equation through a set of algebraic transformations:' Y
Derivation-For the sake of clarity and tractability, this discussion will initially be based on the linear twocompartment model shown in Figure 1A. On the basis of the linear and mammillary structure of this class of models, the conclusions will be generalized for a linear n-compartment model. Consider the most common scenario in modeling the pharmacokinetics of a drug administered intravenously: measurements of plasma drug concentration [C,(t)]a t various times following a rapid intravenous injection of size D are fit to the biexponential equation: 0022-3549/90/0400-0331$0 1.oO/O 0 1990,American Pharmaceutical Association
Journal of Pharmaceutical Sciences I 331 Vol. 79, No. 4, April 7990
where V , (in units of fluid volume) is the apparent volume of the central compartment, k12 and k,, (in units of reciprocal time) are intercompartmental rate constants, and k,, (reciprocal time) is the elimination rate constant. Traditionally, this two-compartment model has been characterized by the following set of linear differential equations:
Note that eq 13 is derived from eq 1and the transformations given by eqs 2-5, without making additional assumptions about the structure of the compartment model. Now, if it is assumed that V2= Vd,, - V,, then the following relationship exists:
(14) Additional insight into the model described by eqs 6 and 7 can be obtained by considering their steady-state behavior following an infinitely long continuous infusion. At steady state, when dX,(t)/dt = dX,(t)/dt = 0, eq 7 can be rewritten as follows:
(7) where X , ( t ) and X J t ) are the amounts of drug in the central and peripheral compartments, respectively, and k,(t) is the rate of drug input as a function of time. Clearly, eq 1,which is written in terms of concentration, is not a solution of eqs 6 and 7,which is written in terms of amount. However, since the concentration of drug in a particular compartment is equal to the amount of drug in the compartment divided by the volume of the compartment, dividing both sides of eq 6 by V , and both sides of eq 7 by V,, the apparent volume of the peripheral compartment, yields:
where C,(t) is the (unmeasurable) apparent concentration of drug in the peripheral compartment. If k,(t) = D W [i.e., k J t ) is an instantaneous bolus of size D given a t time t = 01, it should be possible to solve the differential eqs 8 and 9 to yield eq 1. Indeed, Laplace transforms and eqs 2-5 can be used to show that the solutions of eqs 8 and 9 are as follows:
Cl(t) = Ale-’’‘
+ A2e-Ad
(10)
Equation 10 is identical to eq 1, proving that the model represented by eqs 6 and 7 does, in fact, describe the measured observations (eq 1). It can be shown2 that after a n infinitely long continuous infusion a t a rate of K, eq 1 (and eq 10)determines that the amount of drug in the central and peripheral compartments will be K k , , and ~ k l ~ / ( k , l k l o )respectively, , and that the concentration in the central compartment will be K/(V,k,,). Therefore, if volume of distribution at steady state (Vd,) is defined as follows,
Vd,, = total amount of drug in the body at steady state plasma drug concentration at steady state
(12)
then
(13)
332 I Journal of Pharmaceutical Sciences Vol. 79, No. 4, April 1990
Assuming that Vd,, = V , + V , and the relationship given by eq 14, eq 15 states that in this model the drug concentration in each of the compartments is equal at steady state. Conversely, if one assumes that the steady-state concentrations in the central and peripheral compartments are equal, then V , = k,,V~/k,,(eq 14). Therefore, in the two-compartment model described by eqs 6 and 7, the assumption that V d , = V , + V , is analogous to the assumption that C,(m) = C2(w). On what grounds, though, do we assume that the concentrations in all compartments are equal a t steady state? Intuitively, this is consistent with the notion of passive diffusion through permeable membranes where a concentration gradient could not exist at steady state. By specifying that the steady-state concentration in the peripheral compartment is equal to the steady-state concentration in the central compartment, which in our example is equal to K/(V,k,,), the size of the peripheral compartment is determined asX,(m)/C,(m) = k12Vl/k21, which can lead to absurdly large volumes. Fortunately, the reader appreciates the concept of “apparent” volume. Since the volume and content of the peripheral compartment cannot be measured, it is impossible to determine whether drug in that compartment exists as a small concentration in a large volume or a high concentration in a small volume or so forth. Likewise, the definition of V d , given in eq 12 does not require that it be equal to the s u m of the individual compartment volumes, unless it is again assumed that the concentrations in all compartments are equal at steady state. It is a matter of having n + 2 knowns (A,, A,, A,, A,) and n + 3 unknowns (k,,, k12, k,,, V , , VJ;in this classical construct, the volume of the peripheral compartment is not uniquely defined. If we do not make the assumption that the concentrations in all compartments are necessarily equal at steady state, we observe that the relationships in eq 15 also hold if V, = V,, in which case C,(m) = k,,C,(m)/kzl. “his result is intuitively appealing if one wishes to interpret the ratio kl2:kZ1 a s a partition coefficient. Related to this, the basic differential equations (e.g., eqs 6 and 7) for the compartment model can be written, as some authors have done,3.4 with respect to the concentration of drug in the donor compartment rather than in terms of the amiunt of drug in t h e donor compartment. Doing so, the dynamics of the two-compartment model (Figure 1A) are described by the following:
dCi(t) -dt (16)
vi = EV1
(17)
(22)
then Equations 16 and 17 do require that V, = V,, since they would violate the law of mass balance if V, f V,. The solutions to this system of differential equations are as follows:
Equation 18 is identical to eq 1 (and eq 10) and demonstrates that, from the point of view of what is measured in the plasma (eq l), eqs 16 and 17 and eqs 8 and 9 are equally valid representations of the compartmental structure (Figure 1A).
Conclusions These observations can be summarized by stating there are two equally valid ways of interpreting a linear ncompartment mammillary pharmacokinetic model with central compartment administration, concentration measurement, and elimination (Figure 1B). By one interpretation, that which is most common, Vd, = V, + V, + ... + V,, the concentrations in each of the compartments are equal a t steady state [i.e., C,(m) = C,(m) = ... = C,(=J)]and Vi = l ~ , ~ V , / kfor ~ ,1 = 1, 2, ..., n. In the other interpretations, V, = V , = ... = V,, C,(m) = k , , C , ( = ~ ) / kfor , , i = 1, 2, ..., n, and Vd,, # V, + V, + ... + V, unless Vd,, = nV,. With the definition of Vd, given in eq 12, the following is true in both interpretations:
Computation of the rate constants (k,,, k,,, klo) and central compartment volume (V,)yields the same result irrespective of the model for which they are derived. Actually, there are an infinite number of equally valid ways to interpret the relationship between the peripheral and central compartments. The inviolable relationships can be generalized from eq 15 as follows for i equals 2 to n in the n-compartment model:
(21) When the following relationship exists for i equals 2 to n:
(23) or vice versa. Equation 21 is valid for all values of E > 0, and E cannot be uniquely determined if drug can be measured only from the central compartment. Again, computation of the rate constants and V, is not affected by the choice of E . The cases = kli/k,, and = 1, discussed in this paper were for respectively. There are advantages to the classical development of the compartmental equations exemplified by eqs 6 and 7. First, this approach suggests that the sum of the compartment volumes is equal to the total volume of distribution, which is heuristically pleasing. Second, that the concentrations in each of the compartments are equal a t steady state is consistent with the notion of permeable membranes and passive diffusion of drug molecules. Although there are no particular advantages to the alternative interpretations subsequently presented in this paper, their validity emphasizes that compartmental pharmacokinetic models are only a convenient abstraction of a multiexponential equation. Caution must be exercised when attributing physiological significance to the structure or behavior of these models and, in particular, when attempting to ascribe meaning to the peripheral compartments. Furthermore, other concepts, such as the equality of intercompartmental clearances5 (e.g., k,,V, = k,,VJ may be dependent on one or the other of the interpretations (in this case, the former of the two interpretations detailed above). Compartmental pharmacokinetic modeling is useful in describing and predicting central compartment drug concentrations, but physiological pharmacokinetic modeling6 may offer much more insight into the ways in which drug is actually distributed among the many peripheral compartments or tissues.
References and Notes 1. Cutler, David J. J . Pharmacokinet. Biopharm. 1978,6, 265-282. 2. Wagner, John G . Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications: Hamilton, IL, 1975; pp 82-102. 3. Jacobs, J. R. IEEE Trans. Bwmed. Eng. 1988,35, 763-165. 4. Tavernier, A.; Coussaert, E.; D'Hollander, A.; Cantraine, F. Acta Anaesth. Belg. 1987,38, 63-68. 5. Jusko, William J. In A lied Pharmacokinetics; Evans, W . E.; J., Eds.; Applied Therapeutics: San Schentag, J. J.; Jusko, Francisco, CA, 1980; pp 64-77. 6. Gerlowski, L. E.; Jain, R. K. J . Pharm. Sci. 1983, 72, 1103-1127.
f?
Journal of Pharmaceutical Sciences I 333 Vol. 79, No. 4, April 1990
AND ERICD. HAWKINS' Received May 22, 1989, from the 'Department of Anesthesiology, Duke University Medical Center, Durham, NC 27770, the *Departmentof Anesfhesiol y (7 12A), Stanford University, Veterans Administration Medical Center, 3801 Miranda Avenue, Palo Alto, CA 94304, the SPritzker Institute of %dim1 Engineering, lllinois Institute of Technology, Chicago, IL 60616, and the 'Department of Pharmacology, East Carolina University, School of Medicine, Greenville, NC 27858. Accepted for publication August 8, 1989. __ _ _ _ _ ~ Abstract 0 In pharmacokinetic modeling it is common to use compartmental structures to describe the disposition of a drug in the blood or ~
~
plasma. Typically, a linear multicompartment mammillary model is equated with the multiexponential equation derived from observing the decay of the plasma drug concentration following an intravascular injection.Classically,the mammillary models are constructed so that the concentrationsin each of the compartments areequal at steady state,the apparent volume of distribution at steady state is equal to the sum of the individual Compartment volumes, and the apparent volume of each peripheral compartment is equal to the ratio of its intercompartmental rate constants times the central compartment volume. On the basis of what can be measured in the plasma, however, it is equally valid to assume that the sizes of the peripheral compartment volumes are equal to the central compartment volume and that the steady-state concentration in each peripheral compartment is equal to the ratio of its intercompartmentalrate constants times the Concentration in the central compartment. In fact, these are but two of an infinite number of interpretations of the peripheral Compartment volumes. __ -. .-
A
B
~
Multicompartment models are frequently used to provide an abstraction of the pharmacokinetic behavior of drugs whose disposition in plasma following a rapid intravascular injection can be described by a multiexponential equation. The implications of the compartmental structure are much more intuitively appealing than those of a multiexponential equation. If drug concentration can be measured only in the central compartment, then the volumes of, and the drug concentration in, the peripheral compartments cannot be determined; but, it is generally assumed that the concentration in each of the compartments is equal at steady state. Mathematically, this implies that the steady-state volume of distribution is equal to the sum of the individual compartment volumes. We will show that on the basis of what is measurable (i.e., the concentration of drug in the blood or plasma as a function of dosage and time) it is equally valid to assume that the compartment volumes are equal and that, therefore, the apparent concentration of drug in the compartments a t steady state is proportional to the ratio of the intercompartmental rate constants.
Theoretical Section
Figure 1-(A) Linear two-compartment mammillary pharmacokinetic model; ( 6 )linear n-compartment mammillary pharmacokinetic model.
C,(t) = Ale-A1L + A2e-A2t
(1)
where A, and A , are in units of concentration and A, and A, are in units of reciprocal time. The pharmacokinetics of the drug is assumed to be linear and, on the basis of linear system theory,' eq 1(scaled by D-')represents the impulse response of the system, relating the input (drug dosage) to the output (concentration of drug in the plasma). The biexponential equation has a one-to-one correspondence to a linear two-compartment mammillary model (Figure 1A). where drug administration. measurement. and elimination occur exclusively through the central compartment. The parameters of the two-compartment model are related to the coefficients of the biexponential equation through a set of algebraic transformations:' Y
Derivation-For the sake of clarity and tractability, this discussion will initially be based on the linear twocompartment model shown in Figure 1A. On the basis of the linear and mammillary structure of this class of models, the conclusions will be generalized for a linear n-compartment model. Consider the most common scenario in modeling the pharmacokinetics of a drug administered intravenously: measurements of plasma drug concentration [C,(t)]a t various times following a rapid intravenous injection of size D are fit to the biexponential equation: 0022-3549/90/0400-0331$0 1.oO/O 0 1990,American Pharmaceutical Association
Journal of Pharmaceutical Sciences I 331 Vol. 79, No. 4, April 7990
where V , (in units of fluid volume) is the apparent volume of the central compartment, k12 and k,, (in units of reciprocal time) are intercompartmental rate constants, and k,, (reciprocal time) is the elimination rate constant. Traditionally, this two-compartment model has been characterized by the following set of linear differential equations:
Note that eq 13 is derived from eq 1and the transformations given by eqs 2-5, without making additional assumptions about the structure of the compartment model. Now, if it is assumed that V2= Vd,, - V,, then the following relationship exists:
(14) Additional insight into the model described by eqs 6 and 7 can be obtained by considering their steady-state behavior following an infinitely long continuous infusion. At steady state, when dX,(t)/dt = dX,(t)/dt = 0, eq 7 can be rewritten as follows:
(7) where X , ( t ) and X J t ) are the amounts of drug in the central and peripheral compartments, respectively, and k,(t) is the rate of drug input as a function of time. Clearly, eq 1,which is written in terms of concentration, is not a solution of eqs 6 and 7,which is written in terms of amount. However, since the concentration of drug in a particular compartment is equal to the amount of drug in the compartment divided by the volume of the compartment, dividing both sides of eq 6 by V , and both sides of eq 7 by V,, the apparent volume of the peripheral compartment, yields:
where C,(t) is the (unmeasurable) apparent concentration of drug in the peripheral compartment. If k,(t) = D W [i.e., k J t ) is an instantaneous bolus of size D given a t time t = 01, it should be possible to solve the differential eqs 8 and 9 to yield eq 1. Indeed, Laplace transforms and eqs 2-5 can be used to show that the solutions of eqs 8 and 9 are as follows:
Cl(t) = Ale-’’‘
+ A2e-Ad
(10)
Equation 10 is identical to eq 1, proving that the model represented by eqs 6 and 7 does, in fact, describe the measured observations (eq 1). It can be shown2 that after a n infinitely long continuous infusion a t a rate of K, eq 1 (and eq 10)determines that the amount of drug in the central and peripheral compartments will be K k , , and ~ k l ~ / ( k , l k l o )respectively, , and that the concentration in the central compartment will be K/(V,k,,). Therefore, if volume of distribution at steady state (Vd,) is defined as follows,
Vd,, = total amount of drug in the body at steady state plasma drug concentration at steady state
(12)
then
(13)
332 I Journal of Pharmaceutical Sciences Vol. 79, No. 4, April 1990
Assuming that Vd,, = V , + V , and the relationship given by eq 14, eq 15 states that in this model the drug concentration in each of the compartments is equal at steady state. Conversely, if one assumes that the steady-state concentrations in the central and peripheral compartments are equal, then V , = k,,V~/k,,(eq 14). Therefore, in the two-compartment model described by eqs 6 and 7, the assumption that V d , = V , + V , is analogous to the assumption that C,(m) = C2(w). On what grounds, though, do we assume that the concentrations in all compartments are equal a t steady state? Intuitively, this is consistent with the notion of passive diffusion through permeable membranes where a concentration gradient could not exist at steady state. By specifying that the steady-state concentration in the peripheral compartment is equal to the steady-state concentration in the central compartment, which in our example is equal to K/(V,k,,), the size of the peripheral compartment is determined asX,(m)/C,(m) = k12Vl/k21, which can lead to absurdly large volumes. Fortunately, the reader appreciates the concept of “apparent” volume. Since the volume and content of the peripheral compartment cannot be measured, it is impossible to determine whether drug in that compartment exists as a small concentration in a large volume or a high concentration in a small volume or so forth. Likewise, the definition of V d , given in eq 12 does not require that it be equal to the s u m of the individual compartment volumes, unless it is again assumed that the concentrations in all compartments are equal at steady state. It is a matter of having n + 2 knowns (A,, A,, A,, A,) and n + 3 unknowns (k,,, k12, k,,, V , , VJ;in this classical construct, the volume of the peripheral compartment is not uniquely defined. If we do not make the assumption that the concentrations in all compartments are necessarily equal at steady state, we observe that the relationships in eq 15 also hold if V, = V,, in which case C,(m) = k,,C,(m)/kzl. “his result is intuitively appealing if one wishes to interpret the ratio kl2:kZ1 a s a partition coefficient. Related to this, the basic differential equations (e.g., eqs 6 and 7) for the compartment model can be written, as some authors have done,3.4 with respect to the concentration of drug in the donor compartment rather than in terms of the amiunt of drug in t h e donor compartment. Doing so, the dynamics of the two-compartment model (Figure 1A) are described by the following:
dCi(t) -dt (16)
vi = EV1
(17)
(22)
then Equations 16 and 17 do require that V, = V,, since they would violate the law of mass balance if V, f V,. The solutions to this system of differential equations are as follows:
Equation 18 is identical to eq 1 (and eq 10) and demonstrates that, from the point of view of what is measured in the plasma (eq l), eqs 16 and 17 and eqs 8 and 9 are equally valid representations of the compartmental structure (Figure 1A).
Conclusions These observations can be summarized by stating there are two equally valid ways of interpreting a linear ncompartment mammillary pharmacokinetic model with central compartment administration, concentration measurement, and elimination (Figure 1B). By one interpretation, that which is most common, Vd, = V, + V, + ... + V,, the concentrations in each of the compartments are equal a t steady state [i.e., C,(m) = C,(m) = ... = C,(=J)]and Vi = l ~ , ~ V , / kfor ~ ,1 = 1, 2, ..., n. In the other interpretations, V, = V , = ... = V,, C,(m) = k , , C , ( = ~ ) / kfor , , i = 1, 2, ..., n, and Vd,, # V, + V, + ... + V, unless Vd,, = nV,. With the definition of Vd, given in eq 12, the following is true in both interpretations:
Computation of the rate constants (k,,, k,,, klo) and central compartment volume (V,)yields the same result irrespective of the model for which they are derived. Actually, there are an infinite number of equally valid ways to interpret the relationship between the peripheral and central compartments. The inviolable relationships can be generalized from eq 15 as follows for i equals 2 to n in the n-compartment model:
(21) When the following relationship exists for i equals 2 to n:
(23) or vice versa. Equation 21 is valid for all values of E > 0, and E cannot be uniquely determined if drug can be measured only from the central compartment. Again, computation of the rate constants and V, is not affected by the choice of E . The cases = kli/k,, and = 1, discussed in this paper were for respectively. There are advantages to the classical development of the compartmental equations exemplified by eqs 6 and 7. First, this approach suggests that the sum of the compartment volumes is equal to the total volume of distribution, which is heuristically pleasing. Second, that the concentrations in each of the compartments are equal a t steady state is consistent with the notion of permeable membranes and passive diffusion of drug molecules. Although there are no particular advantages to the alternative interpretations subsequently presented in this paper, their validity emphasizes that compartmental pharmacokinetic models are only a convenient abstraction of a multiexponential equation. Caution must be exercised when attributing physiological significance to the structure or behavior of these models and, in particular, when attempting to ascribe meaning to the peripheral compartments. Furthermore, other concepts, such as the equality of intercompartmental clearances5 (e.g., k,,V, = k,,VJ may be dependent on one or the other of the interpretations (in this case, the former of the two interpretations detailed above). Compartmental pharmacokinetic modeling is useful in describing and predicting central compartment drug concentrations, but physiological pharmacokinetic modeling6 may offer much more insight into the ways in which drug is actually distributed among the many peripheral compartments or tissues.
References and Notes 1. Cutler, David J. J . Pharmacokinet. Biopharm. 1978,6, 265-282. 2. Wagner, John G . Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications: Hamilton, IL, 1975; pp 82-102. 3. Jacobs, J. R. IEEE Trans. Bwmed. Eng. 1988,35, 763-165. 4. Tavernier, A.; Coussaert, E.; D'Hollander, A.; Cantraine, F. Acta Anaesth. Belg. 1987,38, 63-68. 5. Jusko, William J. In A lied Pharmacokinetics; Evans, W . E.; J., Eds.; Applied Therapeutics: San Schentag, J. J.; Jusko, Francisco, CA, 1980; pp 64-77. 6. Gerlowski, L. E.; Jain, R. K. J . Pharm. Sci. 1983, 72, 1103-1127.
f?
Journal of Pharmaceutical Sciences I 333 Vol. 79, No. 4, April 1990