A mathematical analysis of rebound in a target-mediated drug disposition model: I.Without feedback
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 Classification92C45 92C50 34E10 37L25
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).
- Ait-Oudhia S, Lowe P, Mager D (2012) Bridging clinical outcomes of canakinumab treatment in rheumatoid arthritis patients with a population model of il-1\(\beta \) kinetics. Pharmacomet Syst Pharmacol CPT. doi: 10.1038/psp.2012.6
- Aston P, Derks G, Agoram B, van der Graaf P (2012) A mathematical analysis of rebound in a target-mediated drug disposition model. II. With feedback. Abstract at VPH2012. http://www.vph-noe.eu/vph2012
- Bhatia A, Kast R (2007) Tnf can paradoxically increase on etanercept treatment, occasionally contributing to TNF-mediated disease. J Rheum 34:447Google Scholar
- Cacheux W, Boisserie T, Staudacher L, Vignaux O, Dousset B, Soubrane O, Terris B, Mateus C, Chaussade S, Goldwasser F (2008) Reversible tumor growth acceleration following bevacizumab interruption in metastatic colorectal cancer patients scheduled for surgery. Ann Onc 19:1659–1661CrossRefGoogle Scholar
- Chan J, Piper D, Cao Q, Liu D, King C, Wang W, Tang J, Liu Q, Higbee J, Xia A, Di Y, Shetterly S, Arimura Z, Salomonis H, Romanow W, Thibault S, Zhang R, Cao P, Yang X, Yu T, Lu M, Retter M, Kwon G, Henne K, Pan O, Tsai M, Fuchslocher B, Yang E, Zhou L, Lee K, Daris M, Sheng J, Wang Y, Shen W, Yeh W, Emery M, Walker N, Shan B, Schwarz M, Jackson S (2009) A proprotein convertase subtilisin/kexin type 9 neutralizing antibody reduces serum cholesterol in mice and nonhuman primates. Proc Natl Acad Sci USA 106:9820–9825CrossRefGoogle Scholar
- Haringman J, Gerlag D, Smeets T, Baeten D, van den Bosch F, Bresnihan B, Breedveld F, Dinant H, Legay F, Gram H, Loetscher P, Schmouder R, Woodworth T, Tak P (2006) A randomized controlled trial with an anti-ccl2 (anti-monocyte chemotactic protein 1) monoclonal antibody in patients with rheumatoid arthritis. Arthritis Rheum 54:2387–2392CrossRefGoogle Scholar
- Jordan D, Smith P (2007) Nonlinear ordinary differential equations, 4th edn. OUP, New YorkGoogle Scholar
- Klein B, Zhang X, Lu Z, Bataille R (1995) Interleukin-6 in human multiple myeloma. Blood 85:863–872Google Scholar
- Krippendorff D, Oyarzun B-F, Huisinga W (2009) Ligand accumulation counteracts therapeutic inhibition of receptor systems. In: Proceedings of foundations of systems biology in engeneering, FOSBE, pp 173–176Google Scholar
- Peletier L, Gabrielsson J (2012b) Dynamics of target-mediated drug disposition: how a drug reaches its target. Comput Geosc. doi: 10.1007/s10596-012-9312-4
- Smith H (1995) Monotone dynamical systems. In: An introduction to the theory of competitive and cooperative systems, Mathematical surveys and monographs, vol 41. AMS, ProvidenceGoogle Scholar