# More about the Laplace Transform Method

### Student Objectives for this Chapter

- Use the 'finger-print' method to back transform from the Laplace domain
- To understand and use the steps needed to integrate multi-compartment pharmacokinetic models
- Use the convolution method to derive the Laplace of more complex pharmacokinetic models

It is time to explore some more techniques related to the Laplace transform method of integrating differential equations. In an earlier Chapter we saw how we could integrate differential equations using Laplace transforms. At that point we wrote the differential equation, took the Laplace transform of each equation, solved for the model components of interest and used Tables of Laplace transforms to take the back-transform.
In this Chapter we will use the 'finger print' method to take the back transform [1]. We will also extend the method Laplace transforms to integrate differential equations derived from multi-compartment pharmacokinetic models, in the first instance a two compartment model [2]. Finally, we will look at a convolution method to develop the Laplace transform of more complex models [3].

## Finger-Print Method for Inverse Laplace Transform [1, 3]

This method has been described by Benet and Turi [1] and Benet [3]. If the requirements for this method are meet the inverse Laplace transform can be written almost by inspection. The second paper by Benet [3] provides a more complex method where the requirements of no repeating factors in not met.
## Extension to Multi-compartment Pharmacokinetic Models [2]

Using the previous methods and by making judicious substitutions it is possible develop the Laplace equations for models representing two and more compartment models. Rate constants such as k12 and k21 (micro constants) are substituted with macro constants such as α and β. This has been well described by Mayersohn and Gibaldi [2].
## Convolution Method of Deriving Laplace Transforms [3]

Benet [3] has presented a method of developing the Laplace transform for a pharmacokinetic model as the product of the input function and the disposition function. The input function is derived from the route of administration, while the disposition function depends on the complexity of the distribution and elimination processes.

## References

- Benet, L.Z. and Turi, J.S. 1971. "Use of the General Partial Fraction Theorems for Obtaining Inverse Laplace Transforms in Pharmacokinetic Analysis", J. Pharm. Sci., 60: 1593-1594
- Mayersohn, M. and Gibaldi, M 1971 "Mathematical Methods in Pharmacokinetics. II. "Solution of the Two Compartment Open Model," Amer. J. Pharm. Ed., 35:19-28
- Benet, L.Z. 1972 "General Treatment of Linear Mammillary Models with Elimination from any Compartment as Used in Pharmacokinetics," J. Pharm. Sci., 61:536-541

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