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Torus and Homoclinic Bifurcations on a Cells Repair Regulations Model of the Metastatic Nasopharyngeal Carcinoma

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Abstract

Nasopharyngeal carcinoma (NPC) is a tumor that grows on nasopharyngeal epithelial cells. There are some proteins that play important roles in the regulation of cell repair for metastatic NPC, i.e., ATM, p53, MDM2, and DSB. In this paper, we construct a mathematical model that shows the interactions between such proteins and study the behavior of NPC in a long time period. The model is a four-dimensional system of first-order ODE that has a 16-dimensional parameter space. In this case, we consider the appearance of attracting patterns of the solutions near the steady-state conditions. We use the codimension 1 and codimension 2 bifurcations analysis to study the role of several important parameters for the metastasis of NPC and study the possibilities of the system to have chaotic solutions. The appearance of a chaotic solution shows the irregularity of the system due to the changes of the initial conditions. It is important to understand the metastasis behavior NPC and to determine the treatment strategies.

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Acknowledgements

The authors wish to thank Department of Mathematics, Universitas Gadjah Mada and Directorate for the Higher Education, Ministry of Research, Technology, and Higher Education of Indonesia for supporting this research through ”Penelitian Terapan Unggulan Perguruan Tinggi (PDUPT)” UGM 2018 with grant number 200/UN1/DITLIT/DIT-LIT/LT/2018. Some parts of the results are also presented in Wiraya (2017) as a part of the research report which is not being published. The authors also would like to say thanks for some colleagues in UGM and the Cancer Modeling Team UGM for the discussions during the research.

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  1. Ario Wiraya

    Present address: Mathematics Education Study Program, Faculty of Teacher Training and Education, Universitas Sebelas Maret, Surakarta, Indonesia

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  1. Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia

    Ario Wiraya & Fajar Adi-Kusumo

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  1. Ario Wiraya
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Correspondence to Fajar Adi-Kusumo.

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Appendix

Appendix

In Tables 2 and 3, we show the medical interpretation of the variables and parameters of the system, respectively.

Table 2 The medical interpretation of the variables
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Table 3 The medical interpretation of the parameters
Full size table

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Wiraya, A., Adi-Kusumo, F. Torus and Homoclinic Bifurcations on a Cells Repair Regulations Model of the Metastatic Nasopharyngeal Carcinoma. J Nonlinear Sci 33, 60 (2023). https://doi.org/10.1007/s00332-023-09925-x

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