# Geometric singular perturbation analysis of a dynamical target mediated drug disposition model

• Kristian Uldall Kristiansen
Article

## Abstract

In this paper we present a mathematical analysis of a pharmacological ODE model for target mediated drug disposition (TMDD). It is known that solutions of this model undergo four qualitatively different phases. In this paper we provide a mathematical identification of these separate phases by viewing the TMDD model as a singular perturbed system. Our analysis is based on geometric singular perturbation theory and we believe that this approach systemizes—and sheds further light on—the scalings arguments used by previous authors. In particular, we present a novel description of the third phase through a distinguished solution of a nonlinear differential equation. We also describe the solution curve for large values of initial drug doses and recover, en route, a result by Aston et al. (J Math Biol 68(6):1453–1478, 2014) on rebounding using our alternative perturbation approach. Finally, from our main result we derive a new method for estimating the parameters of the system in the event that detailed data is available. Ideally our approach to the TMDD model should stimulate further research into applications of these methods to more complicated models in pharmacology.

## Keywords

Geometric singular perturbation theory pharmacology target mediated drug disposition

## Notes

### Acknowledgements

The author would like to thank the students Anders Eltved, Sigrun Nordli and Asger Limkilde for their initial work on this problem

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