Nonlinear dynamics and control in a tumor-immune system
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Advances in modeling tumor-immune dynamics and therapies offer deeper understandings of the mechanism of tumor evolution in the interdisciplinary field of mathematics and immune-oncology. The main mathematical models are constructed in terms of ordinary differential equations (ODEs) or partial differential equations (PDEs) and analyzed through tools such as Poincaré map, simulation, or numerical bifurcation analysis to understand the system properties. These models succeed in characterizing essential features of tumor behaviors including periodic bursts and the existence of latency. In relationship to practice, these models are also applied to estimate the feasibility and efficacy of treatments ranging from traditional chemotherapy to immunotherapy (ACI). In recent literature, there have been applications of control methods such as optimal control, hybrid automata, and feedback linearization-based tracking control with almost disturbance decoupling in the studies of tumor-immune systems. This thesis presents an attempt to apply the bifurcation control method with washout filters in tumor treatments. This thesis research investigates the dynamics and controlling of the tumor-immune response of immunotherapies, mainly the Adoptive Cell Immunotherapy (ACI) and Interleukin-2 (IL-2). The first part of the thesis presents the nonlinear dynamics of the classic nonlinear ODE tumor-immune model given by Denise Kirschner and John Carl Panetta in 1998. This model concentrates on the nonlinear phenomena of the tumor-immune system under immunotherapies, primarily the bifurcation phenomenon along with the antigenicity of effector cells. Bifurcation phenomena refer to the qualitative changes in system dynamics due to quasi-static changes in system parameters. Antigenicity refers to a capability to distinguish tumor cells from healthy cells. The Kirschner-Panetta model captures a saddle-node bifurcation and a Hopf bifurcation of the tumor-immune response, which separates the tumor evolution into three stages, the “dangerous equilibrium”, the periodic recurrence, and the “safe equilibrium”. The second part applies and analyzes several control strategies on the immunotherapies based on the KP model in order to eradicate tumors or inhibit tumor growths. The first section studies the combination immunotherapy of ACI and IL-2 as an open-loop control system based on Kirschner’s work, which generates a locally asymptotically stable equilibrium. In the second section, this thesis provides a new idea of treatment in the tumor-immune system, that is a closed-loop control strategy taking advantage of its bifurcation structure by applying dynamic feedback control with a washout filter of ACI or IL-2. Bifurcation control moves the Hopf bifurcation point without changing the equilibrium structure as the bifurcation parameter varies. In this tumor-immune case, the linear dynamic feedback control with a washout filter of ACI could either extend the “safe equilibrium” region or reduce the amplitude of the tumor population at the stage of tumor recurrence. In addition, other bifurcation amplitude controls of either ACI or IL-2 are attempted to reduce the amplitudes of periodic orbits of the tumor immune system but without obvious effects.
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