Pharmacokinetics Laplace transformation

When we apply these two properties of the Laplace transform to differential equations of our pharmacokinetic model in eq. (39.46), we obtain ... 

This general approach for solving linear pharmacokinetic problems is referred to as the y-method. It is a generalization of the approach by means of the Laplace transform, which has been applied in the previous Section 39.1.6 to the case of a two-compartment model. 

F. F. Farris, R.L. Dedrick, P.V. Allen and J.C. Smith, Physiological model for the pharmacokinetics of methyl mercury in the growing rat. Toxicol. Appl. Pharmacol., 119 (1993) 74—90. P. Franklin, An Introduction to Fourier Methods and the Laplace Transformation. Dover, New York, 1958. 

Laplace transformation is particularly useful in pharmacokinetics where a number of series first-order reactions are used to model the kinetics of drug absorption, distribution, metabolism, and excretion. Likewise, the relaxation kinetics of certain multistep chemical and physical processes are well suited for the use of Laplace transforms. 

Many drugs undergo complex in vitro drug degradations and biotransformations in the body (i.e., pharmacokinetics). The approaches to solve the rate equations described so far (i.e., analytical method) cannot handle complex rate processes without some difficulty. The Laplace transform method is a simple method for solving ordinary linear differential equations. Although the Laplace transform method has been used for more complex applications in physics, engineering, and other research areas, here it will be applied to ordinary differential equations of first-order rate processes. 

In chemical degradation kinetics and pharmacokinetics, the methods of eigenvalue and Laplace transform have been employed for complex systems, and a choice between two methods is up to the individual and dependent upon the algebraic steps required to obtain the final solution. The eigenvalue method and the Laplace transform method derive the general solution from various possible cases, and then the specific case is applied to the general solution. When the specific problem is complicated, the Laplace transform method is easy to use. The reversible and consecutive series reactions described in Section 5.6 can be easily solved by the Laplace transform method ... 

Schalla, M. and Weiss, M., Pharmacokinetic curve fitting using numerical inverse Laplace transformation, European Journal of Pharmaceutical Sciences, Vol. 7, 1999, pp. 305-309. 

We have selected a simple example to illustrate the use of Laplace transform methods. A more advanced application is given in the next chapter, in which equations are derived for a two-compartment model. It will be shown subsequently that Laplace transform methods also are helpful in pharmacokinetics when convolution/deconvolution methods are used to characterize drug absorption processes. 

Mayersohn, M. Gibaldi, M. Mathematical methods in pharmacokinetics. I. Use of the laplace transform for solving differential rate equations. Am. J. Pharm. Ed. 1970, 34, 608-614. 

Benet, L.Z. Turi, J.S. Use of general partial fraction theorem for obtaining inverse laplace transforms in pharmacokinetic analysis. J. Pharm. Sci. 1971, 60, 1593-1594. 

This technique will be useful In the pharmacokinetic evaluation of drugs which cannot be given by a single quick bolus injection, because of potential toxicity, irritation or limited solubility. These authors utilized Laplace, transform input functions In deriving their equations, a rapid and easy method for deriving pharmacokinetic equations which should find expanded use in the next year. 

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