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

Urszula Ledzewicz; Omeiza Olumoye; Heinz Schättler

doi:10.3934/mbe.2013.10.787

Article Contents

2013, Volume 10, Issue 3: 787-802. Doi: 10.3934/mbe.2013.10.787

This issue Previous Article Equilibrium solutions for microscopic stochastic systems in population dynamics Next Article On the MTD paradigm and optimal control for multi-drug cancer chemotherapy

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

Received: August 31, 2012

Revised: December 31, 2012

Published: March 2013

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Abstract

In this paper, a mathematical model for chemotherapy that takes tumor immune-system interactions into account is considered for a strongly targeted agent. We use a classical model originally formulated by Stepanova, but replace exponential tumor growth with a generalised 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 the limit $\nu\rightarrow\infty$). The dynamics is multi-stable and equilibria and their stability will be investigated depending on the parameter $\nu$. Except for small values of $\nu$, the system has both an asymptotically stable microscopic (benign) equilibrium point and an asymptotically stable macroscopic (malignant) equilibrium point. The corresponding regions of attraction are separated by the stable manifold of a saddle. The optimal control problem of moving an initial condition that lies in the malignant region into the benign region is formulated and the structure of optimal singular controls is determined.

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Mathematics Subject Classification: Primary: 49K15, 92C50; Secondary: 93C95.

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