Dynamical Control for the Parametric Uncertain Cancer Systems
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Dynamical Control for the Parametric Uncertain Cancer Systems Yi-Horng Lai, Lan-Yuen Guo, Kun-Ching Wang, and Jau-Woei Perng* Abstract: In this study, we consider a parametric uncertain Lotka–Volterra cancer model including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. The biological parameter (i.e., cell growth rate) is described as a form of the triangular fuzzy number. By using grade mean value conversion, the imprecise fuzzy parameter is translated into the degree of optimism (λ -integral value λ ∈ [0, 1]) interval. We derive the sufficient conditions for the existence of the region of asymptotic stability (RAS) in the fuzzy cancer model. The boundary crisis of transient chaos and properties of RAS are investigated under fuzzy environment. We present a dynamical perturbation control to avoid uncontrolled tumor cell growth and prevent healthy cell extinction. Keywords: Cancer model, Lotka-Volterra system, parametric uncertain system, transient chaos.
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INTRODUCTION
Cancer is an abnormal growth of cells in the tissues. Mathematical models have been applied to investigate the properties of cancer systems. The general Lotka–Volterra models are an appropriate starting point for cancer systems. Applying the Lotka–Volterra model to evaluate the interaction between tumor cells, healthy host cells, and immune cells has been addressed in [1, 2]. Lettellier et al. adopted a topological method to analyze the bifurcation phenomena of the cancer model [3]. Global asymptotic behavior and equilibrium point analysis for the cancer model was examined in [4]. Dong et al. explored and illustrated how time delays affect the stability region in cancer systems [5]. Chaotic dynamics can also be observed through equilibria analysis in the Lotka–Volterra cancer model [6]. Moreover, transient chaotic dynamics of the cancer model were investigated in [7]. Surgery, radiotherapy, and chemotherapy all play key roles in the treatment of cancer. In addition, many novel varieties of cancer treatments are currently available. One of the most significant advances in cancer treatment is the development of adoptive cellular immunotherapy (ACI) [8]. ACI refers to the injection of the patient’s own immune cells directly into the tumor-bearing host to boost the patient’s immune reaction [9]. In order to investigate cancer progression under differ-
ent treatments, the treatment term has to be added into the cancer model. De Pillis et al. proposed an optimally controlled chemotherapy model for comparison with traditional pulsed chemotherapy [1, 10]. In [11], the mixed chemoimmunotherapy model was validated by applying mouse and human parameters. Such treatment models usually analyze the properties of equilibrium points and the bifurcation phenomenon, the chaotic behavior is seldom addressed in these control models. Appling fuzzy sampled-data controller to synchronize the uncertain chaotic systems was investigated in [12, 13]. In [14, 15]
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