# A mathematical analysis of rebound in a target-mediated drug disposition model: I.Without feedback

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## Abstract

We consider the possibility of free receptor (antigen/cytokine) levels rebounding to higher than the baseline level after one or more applications of an antibody drug using a target-mediated drug disposition model. Using geometry and dynamical systems analysis, we show that rebound will occur if and only if the elimination rate of the drug–receptor product is slower than the elimination rates of the drug and of the receptor. We also analyse the magnitude of rebound through approximations and simulations and demonstrate that it increases if the drug dose increases or if the difference between the elimination rate of the drug–receptor product and the minimum of the elimination rates of the drug and of the receptor increases.

## Mathematics Subject Classification

92C45 92C50 34E10 37L25## Notes

### Acknowledgments

We are grateful for the input of Adewale Raji in the preliminary work for this paper. This preliminary stage was in part financially supported through the Biopharma Skills Project of the Universities of Surrey and Reading, jointly funded by the Higher Education Funding Council for England’s Economic Challenge Investment Fund (ECIF) and the South East England Development Agency (SEEDA).

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