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On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth

1. Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653
2. Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130

## Abstract    Related pages

In this paper, a mathematical model for chemotherapy that takestumor immune-system interactions into account is considered for astrongly targeted agent. We use a classical model originallyformulated by Stepanova, but replace exponential tumor growth with ageneralised logistic growth model function depending on a parameter$\nu$. This growth function interpolates between a Gompertzian model(in the limit $\nu\rightarrow0$) and an exponential model (in thelimit $\nu\rightarrow\infty$). The dynamics is multi-stable andequilibria and their stability will be investigated depending on theparameter $\nu$. Except for small values of $\nu$, the system hasboth an asymptotically stable microscopic (benign) equilibrium pointand an asymptotically stable macroscopic (malignant) equilibriumpoint. The corresponding regions of attraction are separated by thestable manifold of a saddle. The optimal control problem of movingan initial condition that lies in the malignant region into thebenign region is formulated and the structure of optimal singularcontrols is determined.
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Citation: Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences and Engineering, 2013, 10(3): 787-802. doi: 10.3934/mbe.2013.10.787

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• 1. U. Ledzewicz, H. Schättler, S. Anita, N. Hritonenko, G. Marinoschi, A. Swierniak, A Review of Optimal Chemotherapy Protocols: From MTD towards Metronomic Therapy, Mathematical Modelling of Natural Phenomena, 2014, 9, 4, 131, 10.1051/mmnp/20149409
• 2. Heinz Schättler, Urszula Ledzewicz, Behrooz Amini, Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy, Journal of Mathematical Biology, 2016, 72, 5, 1255, 10.1007/s00285-015-0907-y
• 3. Gary An, Swati Kulkarni, An agent-based modeling framework linking inflammation and cancer using evolutionary principles: Description of a generative hierarchy for the hallmarks of cancer and developing a bridge between mechanism and epidemiological data, Mathematical Biosciences, 2015, 260, 16, 10.1016/j.mbs.2014.07.009
• 4. Urszula Ledzewicz, Behrooz Amini, Heinz Schättler, Dynamics and control of a mathematical model for metronomic chemotherapy, Mathematical Biosciences and Engineering, 2015, 12, 6, 1257, 10.3934/mbe.2015.12.1257
• 5. Nicolas Houy, François Le Grand, Francesco Pappalardo, Optimal dynamic regimens with artificial intelligence: The case of temozolomide, PLOS ONE, 2018, 13, 6, e0199076, 10.1371/journal.pone.0199076
• 6. Dominique Barbolosi, Joseph Ciccolini, Bruno Lacarelle, Fabrice Barlési, Nicolas André, Computational oncology — mathematical modelling of drug regimens for precision medicine, Nature Reviews Clinical Oncology, 2016, 13, 4, 242, 10.1038/nrclinonc.2015.204
• 7. Heinz Schättler, Urszula Ledzewicz, , Optimal Control for Mathematical Models of Cancer Therapies, 2015, Chapter 8, 317, 10.1007/978-1-4939-2972-6_8
• 8. Heinz Schättler, Urszula Ledzewicz, , Optimal Control for Mathematical Models of Cancer Therapies, 2015, Chapter 1, 1, 10.1007/978-1-4939-2972-6_1
• 9. Urszula Ledzewicz, Heinz Schättler, , Mathematical Models of Tumor-Immune System Dynamics, 2014, Chapter 7, 157, 10.1007/978-1-4939-1793-8_7