APPLICATION OF LAPLACE TRANSFORMS TO A PHARMACOKINETIC OPEN TWO-COMPARTMENT BODY MODEL

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1 Acta Poloniae Pharmaceutica ñ Drug Research, Vol. 74 No. 5 pp. 1527ñ1531, 2017 ISSN Polish Pharmaceutical Society APPLICATION OF LAPLACE TRANSFORMS TO A PHARMACOKINETIC OPEN TWO-COMPARTMENT BODY MODEL TADEUSZ W ADYS AW HERMANN Department of Physical Pharmacy and Pharmacokinetics, PoznaÒ University of Medical Sciences, 6 åwiícickiego St., PoznaÒ, Poland Abstract: The anti-laplace of complicated transforms for two-compartment differential equations may be found only in an extensive table of Laplace transforms which usually are not available. Therefore, a general partial fraction theorem was used for obtaining their inverse Laplace transforms. First, the disposition function of the central compartment in a linear two-compartment mammillary model, d s,c must be written down. Second, except for that disposition function appropriate intake functions were considered with intravenous instantaneous bolus injection, constant zero-order rate infusion, and intramuscular first-order single dose injection of indobufen (Ibustrin). The product of the disposition function and the appropriate input function yields the experimental Laplace transforms for the amount (concentrations) of drug in the central compartment, a s,c (X C ). The input functions for the above mentioned routes of drug administration are equal: X 0 (the dose), k 0 (1 ñ e ñt s )/S, and k a F X 0 / (s + k a ), respectively. The equations derived are also illustrated in four figures. Keywords: two-compartment body model, Laplace transforms, the inverse Laplace, partial fraction theorem, disposition function, different intake function: bolus injection, infusion, intramuscular injection The use of Laplace transforms of a function, LF(t), and the inverse operation, L -1, may be applied to obtain solutions to the systems of linear differential equations of the first-order. However, sufficient matrix algebra is needed to use the above transforms (1). That operation transforms differentiation into multiplication as well as integration into division (1). The Laplace transform enables complex rate expressions to be manipulated easily by conventional algebraic techniques (2). However, it is true with regards to a one-compartment body model. The anti- Laplace of the resulting complicated transforms may be found only in an extensive table of Laplace transforms if two-compartment body model is concerned. Therefore, according to Gibaldi and Perrier (2), and Benet (3, 4), it is easier to use a general partial fraction theorem for obtaining inverse Laplace transforms (4). To illustrate the application of this approach for solving linear differential equations of the first-order a two-compartment pharmacokinetic model with the first- or zero-, or bolus (instantaneous) input will be employed. It should be mentioned that the application of the Laplace transforms to a one-compartment body model is much easier and has been explained quite simply (2, 5, 6). A two-compartment model with intravenous bolus injection The experimental Laplace transform for the disposition function of the central compartment, d s,c in a linear two-compartment mammillary model where elimination of drug from central compartment occurs has been provided (2-4): d s,c = ññññññññññññ (1) where: E 1 = k 10 + and E 2 =, and λ i is a disposition rate constant, which may be expressed in terms of the above individual intercompartmental transfer rate constants and elimination rate constants, Π ñ continued product, s ñ the Laplace operator which replaces the time domain of a rate equation. The model concerned is presented below: X c X T where X C and X T are the amounts of a drug as a function of time in central and tissue compartments, and,, and k 10 are the first-order rate constants of * Corresponding author: hermann@ump.edu.pl; phone:

2 1528 TADEUSZ W ADYS AW HERMANN transfer a drug from the central compartment to the tissue compartment and vice versa as well as elimination rate constant from the central compartment, respectively. The product of the input, in s, and disposition functions yields the Laplace transform for the amount of drug in the central compartment, a s.c a s,c = in s d s,c (2) where in s = X 0 (a single dose injected) for the above model. Therefore a s,c = X 0 ññññññññññññ (3) The above equation may be rewritten for the central compartment amounts of drug following intravenous bolus injection X 0 (s + E 2 ) P(s) a s,c = ññññññññññññññññ = ñññññññññ (4) (s + λ 1 ) (s + λ 2 ) Q(s) The anti-laplace of the resulting transform, L -1 may be obtained by use of a general partial fraction theorem. If the quotient of two polynomials P(s)/Q(s) is given as above, then P(s) P(λ i ) L -1 ññññññ = Σ N ñññññññññ eλ t i (5) Q(s) Q i (λ i ) The roots of the polynomial, Q(s) are λ 1 = ñ α and λ 2 = ñ β. The term Q i (λ i ) may be defined as follows. When i = 1 Q i (λ i ) = (λ i + β) = (β ñ α) i = 2 Q i (λ i ) = (λ 2 + α) = () (6) Hence the solution for the amount of drug in the central compartment, X C, applying the general partial fraction equation, is obtained (E i ñ λ l ) X C = X 0 Σ N ññññññññññ l=1 eλ t i = (λ i ñ λ l ) X 0 (K 21 ñ α) X 0 (K 21 ñ β) = ññññññññññ e ñα t + ññññññññññ e ñβ t (β ñ α) () X 0 XC = ññññññ [(α ñ ) e ñα t + ( ñ β) e ñβ t X 0 (α ñ ) X 0 ( ñ β) when ññññññññññññ = A i ñññññññññññññ = B (7) X C = A e ñα t + B e ñβ t (8) A plot of the logarithms of drug plasma concentrations (C = X C /V d ) versus time according to the Figure 1. Semilog plot of plasma levels vs. time after intravenous bolus administration of the iodine salt to a healthy volunteer described by biexponential equation and the method of residuals (2, 6)

3 Application of Laplace transforms to a pharmacokinetic Figure 2. Serum concentrations of (-)-R-indobufen enantiomer vs. time after single 200 mg intramuscular dose administration of racemic indobufen (Ibustrin) to a healthy volunteer described by TopFit 2.0 computer program for an open two-compartment body model (6, 7) above equation will yield a biexponential curve (Fig. 1). The disposition constants α and β may be obtained applying the method of residuals (Fig. 1). A two-compartment model with a first-order input process For a drug that enters the body by an apparent first-order absorption process (generally via the oral or intramuscular routes) and distributes in the body according to a two-compartment model: X a k a X C X T where: X a ñ amount of drug at an extravascular site of absorption as a function of time, k a ñ the firstorder rate constant of absorption. The disposition function for the central compartment is identical to the disposition function for an intravenous bolus injection d s,c = ññññññññññññ (9) However, the input function is different to describe first-order absorption (2, 5, 6) k a F X 0 ins = ñññññññññññññññ (10) s + k a where F determines the fraction of drug absorbed. The Laplace transform for the amount of drug in the central compartment equals the product of the disposition and input functions k a F X 0 a s,c = ñññññññññññññññññññ (11) (s + k a ) Solving for the amount of drug in the central compartment by taking anti-laplace yields (E i ñ k a ) X C k a F X 0 ñññññññññññññññññññ e ñk a t + (λ i ñ k a ) + k a F X 0 Σ 2 (E i ñ λ l ) ññññññññññññññññññ eñλ l t (12) (k a ñ λ l ) (λ i ñ λ l ) ñ k a XC = k a F X 0 ñññññññññññññññññññ e ñk a t + (λ 1 ñ k a ) (λ 2 ñ k a ) ñ λ + 1 ka F X 0 ñññññññññññññññññññ e ñλ 1 t + (k a ñ λ 1 ) (λ 1 ñ λ 2 ) ñ λ + 2 ka F X 0 ñññññññññññññññññññ e ñλ 2 (k t a ñ λ 2 ) (λ 2 ñ λ 1 ) When the hybrid rate constants α and β (1) as well as the other constants B, A, and C 0 ñ the corresponding zero-time intercepts obtained by the method of residuals, are substituted, respectively, the equation may be transformed to a simpler form: ñ k a XC k a F X 0 [ ñññññññññññññññññññ e ñk a t + (α ñ k a ) (β ñ k a ) ñ α ñ β + ññññññññññññññ e ñα t + ñññññññññññññññ e (k ñβ t a ñ α) (β ñ α) (k a ñ β) () X C = B e ñβ t + A e ñα t ñ C 0 e ñk a t (13) The plots of the above equation presented on both a graph and on a semilog papers are very char-

4 1530 TADEUSZ W ADYS AW HERMANN Figure 3. Semilog plot of serum (-)-R-indobufen enantiomer vs. time after single 200 mg intramuscular dose administration of racemic indobufen (Ibustrin) to a healthy volunteer described by TopFit 2.0 computer program for an open two-compartment body model (6, 7) Figure 4. Semilog plot of plasma levels vs. time of a drug that confers two-compartment model characteristics, following constant rate intravenous infusion to steady-state ( ) and following the rapid intravenous injection of a dose that gives an initial drug concentration equal to the steady-state concentration ( _ ) (2)

5 Application of Laplace transforms to a pharmacokinetic acteristic for a two-compartment model (Fig. 2 and 3). The experimental data presented in Fig. 2 and 3 have been excerpted from a published article (7). A two-compartment model for an intravenous infusion k 0 X C X T where: k 0 ñ the zero-order rate of drug infusion (constant). The disposition function for the central compartment is identical to the disposition function for an intravenous bolus injection. s + E 2 ds,c = ññññññññññññññññññ (14) (s + λ 1 ) (s + λ 2 ) Multiplication of this disposition function by the input function for an intravenous infusion beginning at time zero (i.e., in s = k 0 (1 ñ e -T s )/s (2, 5, 6) yields the following Laplace transform for the amount of drug in the central compartment k 0 (s + E 2 ) (1 ñ e -T s ) a s,c = ññññññññññññññññññ (15) s (s + λ 1 ) (s + λ 2 ) where T is the duration of an infusion. The above two polynomials fulfill the requirements for the use of a partial fraction theorem for obtaining inverse Laplace transforms. Hence the solution for the amount of drug in the central compartment X C as a function of time may be written k 0 (E 2 ñ λ 1 ) (1 ñ e -λ l T ) X c = ññññññññññññññññññññññ e ñλ l t + ñ λ 1 (λ 2 ñ λ 1 ) k 0 (E 2 ñ λ 2 ) (1 ñ e -λ 2 T ) + ññññññññññññññññññññññ e ñλ 2 t (16) ñ λ 2 (λ 1 ñ λ 2 ) The roots of the polynomial, Q(s) were specified previously, and the equation may be rewritten k 0 (α ñ ) (1 ñ e -λ l T ) X c = ññññññññññññññññññññññ e ñα t + α () k 0 (β ñ ) (1 ñ e -β T ) + ññññññññññññññññññññññ e ñβ t (17) β (β ñ α) where: k 0 is the zero-order rate of the infusion. It should be noted that the above single equation describes the amount of drug in the central compartment as a function of time while the infusion is being carried out and after infusion stops. While the infusion is continuing, T = t and varies with time k 0 (α ñ ) (e -ñα t ñ e -ñ2α t ) X c = ññññññññññññññññññññññ + α () k 0 (β ñ ) (e -ñβ t ñ e -ñ2β t ) + ññññññññññññññññññññññ (18) β (β ñ α) However, when infusion ceases, time t becomes a constant corresponding to T ñ the duration of the infusion. It should be apparent that upon stopping the infusion, drug concentrations in the plasma decline in a biexponential manner (Fig. 4). It can be seen from Fig. 4 that the biexponential characteristics of the drug is more evident following the bolus injection than after terminating the infusion. Acknowledgment This publication has been written to give thanks to the Almighty God for my 80 th birthday anniversary which hopefully I am supposed to survive on June 13, REFERENCES 1. Wagner J.G., Pernarowski M.: Biopharmaceutics and relevant pharmacokinetics, Drug Intelligence Publications, Hamilton, Illinois Gibaldi M., Perrier D.: Pharmacokinetics, 2 nd edn., Marcel Dekker, New York and Basel Benet L.Z.: J. Pharm. Sci. 60, 1593 (1971). 4. Benet L.Z.: J. Pharm. Sci. 61, 536 (1972). 5. Hermann T.W.: Farm. Pol. 71, 289 (2015). 6. Hermann T.W.: Pharmacokinetics. Theory and practice. PZWL, Warszawa G Ûwka F.K.: Chirality 12, 38 (2000). Received: