Exp Toxic Patho11999; 51: 409-411 URBAN & FISCHER http://www.urbanfischer.de/joumals/exptoxpath

Department of Pharmacology, Medical faculty, University of Novi Sad, Yugoslavia

Derivation of Laplace transform for the general disposition deconvolution equation in drug metabolism kinetics

JOVAN POPOVIC

With 1 figure

Address for correspondence: Prof. Dr. JOVAN Popovr<:, Department of Pharmacology, Faculty of Medicine, University of Novi Sad, 21000 - Novi Sad, Hajduk Veljkova 3, P.o. Box 380, FR Yugoslavia; fax: ++ 381 (21) 615-771.

Key words: Drug metabolism, kinetics; Laplace transform; Deconvolution equation; Disposition function.

Summary

This paper presents some very simplified general treat­ments which will allow workers to derive equations for any linear kinetic metabolic process. This is done through the use of the Laplace transforms. A general equation is pre­sented to describe the disposition function in Laplace ope­rators for the central compartment of a linear n compartment mammillary model with elimination occurring from any of the compartments. Input functions describes IV bolus, zero­order infusion, intramuscular injection or GI absorption. The Laplace transform for the amount of drug or metabolite in the central compartment is given by the products of the input and disposition functions.

Introduction

In this work, a disposition function defines the model necessary to describe accurately drug body concentrations after the drug has entered the blood circulation. That is, disposition describes everything that happens to a drug, i.e. distribution, metabolism and undirectional elimination through all possible routes, as if all input into the circula­tion, occurred instantaneously. Input functions describe the process necessary to get the dose into the bloodstream. They may either describe an intravenous bolus injection, an intravenous infusion, a first- or zero-order absorption from a site such as the gut or muscle, or any combination of these methods of input. The products of the Laplace transform for the input function and the Laplace trans­form for the disposition function yield the Laplace trans­form for the equation describing the time course of the amount of drug in a compartment. A general model de­scribing input and disposition function is presented in fi­gure 1.

3

j

urin n ..

Fig. 1. The disposition function describes the mammillary model necessary to describe accurately drug body concen­tration after the drug has gotten into the blood circulation.

Methods

Mathematical model:

~il = -kllql + k21 q2 + k31q3 + ... + kn1qn

dq2_k k dt - 12ql - 22q2

dq3-k k dt - 13ql + 33q3

0940-2993/99/51/04-05-409 $ 12.00/0 409

ql - amount of drug or metabolite in central compartment q2' q3' ... , qn - amount of drug/metabolite in peripherial com­

partments kij - first-order rate constants kii - sum of the exit rate constants out of compartment i.

Initial conditions:

ql(O) = D, qiO) = q3(0) = ... = qn(O) = 0

Result

F ()[( k) k21k12 k3lkI3 knlkln]_ D IS s+ 11 - k - k _ ... - lr -s+ 22 S+ 33 S+Ann

Laplace transform for the mathematical model: Lf(t) = F(s),

L df(t) = sF(s) - f(O) dt

sF/s) - D = -k11FI(s) + k21 F2(s) + k3IF3(s) + ... knIFn(s)

sFis) = k12FI(s) - k22Fis) ~ F2(S) = k12Fkl(S) s + 22

sFls) = kI3FI(s) - k33F3(S) ~ F3(S) = kI3Fkl(S) : S+ n

sFn(s) = klnFI(s) - knnFn(s) ~ Fn(s) = klnF~(S) S + Ann

[

(S + k11 )(s + k22)(s + k33)·· ·(S + knn) - k21kuCs + k33)·· ·(s + knn) - k3lkI3 fX (s + kJ - ... - knlkln 8 (s + ki)] FI(s) = D / 1",3 I"'n

(S + k22)(s + k33)·· ·(S + knn)

D IT (s + kJ FI(s) = n 1~2

IT (s + k) - t klkl fI (s + k) i~1 11 j~2 J J i~1 1

i:;t:j

FI(s) / D = a general disposition equation (W).

Discussion

The following input functions, in, pictured in figure 1, describe the usual methods utilized to get drug into the central compartment.

For an intravenous bolus:

in(s) = Dose,

where s = an auxiliary variable introduced with the Laplace transforms.

For intravenous infusion or zero-order absorption:

in(s) = ko (e-aS - e-bs) / s,

where ko = zero-order infusion rate in units of amount per time, a = time when infusion begins and b = time when infusion ends.

For first-order absorption:

in(s) = ka Dose / (s + ka),

where ka = first-order absorption rate constant.

The relation between the input function, the disposition function and the amount of drug in a compartment is a convolution integral:

t

q(t) = f in (x) W (t - x) dx = in(t) * Wet). o

But, the relation between the Laplace transform for the input function, the Laplace transform for the disposition

410 Exp Toxic Patho151 (1999) 4-5

function and the Laplace transform for the equation de­scribing the time course of the amount of drug in a compart­ment q(s) is, according to the convolution theorem:

q(s) = in(s) . W(s).

The Laplace transform for the amount of drug in a com­partment is given by the product of the Laplace transform for the input function and the Laplace transform for the disposition function.

The deconvolution, which means the reverse calcula­tion of the convolution is often useful for linear compart­ment analysis in pharmacokinetic studies. Drug absorption rate process from pharmaceutical preparations, metabolic rate process in a whole body, and elimination rate process of drug metabolites, etc. can be analyzed without assump­tion of proper kinetic models by the application of decon­volution calculation.

The anti-Laplace could be found in an extensive table of Laplace transforms, but the general method of partial fractions is much easier and is applicable in the majority of cases.

If the quotient of two polynomials, P( s )/Q( s), is such that Q(s) has a higher degree and contains the factor (S-AJ, which is not repeated, then:

where: A/ s = roots of the polynomial Q(s), Q'(Ai) = derivative of Q(s) with the roots substituted for s, Qi(A;) = value of the denominator when Ai is substituted

for all the s terms except for the term originally containing Ai' this term being omitted,

s = the standard notation used in Laplace operations.

The first sum is often called Heaviside' s expansion and the second sum is the general partial fraction theorem.

When repeated functions appear in the denominator of a Laplace transform, Heaviside's expansion and the ge­neral partial fraction theorem, may not be used. Consider the following general equation:

pes) pes) Q(s) = (s - A[),+I (s - A2)(S - A3)" ·(s - Am)

where: pes) = a polynomial in s such that Q(s) has the higher de­

gree in s, (S-AI) = a repeated function in Q(s), that is, r has a value

greater than zero.

The solution to this equation is given by:

L-I (P(S)} = l ~ [<p(s)est] +! peA;) eMt Q(s) r! as' i=2 Q;(A;)

s = Al

where <P(s) is set equal to P(s)lQ(s) when the repeating function is omitted from Q(s). Therefore:

pes) <p(s) Q(s) = (s - AI),+I

Thus, the first term in the solution is the derivative of <P(s)est with respect to s evaluated at s = AI' The second term is similar to the general partial fraction theorem, ex-

cept that the function is not evaluated at AI' the root cor­responding to the repeated function.

Conclusion

The general methods presented in this paper should markedly lessen the amount of work necessary in deriving metabolism kinetic equations folloving linear kinetics. The use of input and disposition functions was suggested, and a general equation to describe the disposition func­tion for the central compartment of a mammillary model, with elimination allowed from any compartment was pre­sented. A general partial fraction theorem which allows the researcher to solve Laplace transforms in a single step was presented. By using the general methods presented in this paper, most drug metabolism equations for multi­compartment models can be solved in four or five simple steps.

Acknowledgement: The excellent technical assistance of Mrs. VESNA POPOVIC is gratefully acknowledged.

References

POPOVIC J: Comparison between classical and general treat­ment of linear pharmacokinetics. Acta BioI Med Exp 1996; 21: 17-20.

POPOVIC J: Polynomials vs cubic spline functions for model independent deconvolution calculations of absorption rate. Eur J Clin Pharmacol1997; 52: 446.

POPOVIC J, MIKOV M, JAKOVLJEVIC V: Pharmacokinetics of carbamazepine derived from a new tablet formulation. Eur J Drug Metab Pharmacokinet 1995; 20: 297-300.

POPOVIC J: Use of Heaviside's expansion theorem for ob­taining inverse Laplace transforms in pharmacokinetic analysis. Acta BioI Med Exp 1996; 21: 53.

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