# American Institute of Mathematical Sciences

February  2004, 4(1): 39-58. doi: 10.3934/dcdsb.2004.4.39

## A mathematical model of tumor-immune evasion and siRNA treatment

 1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan, United States, United States 2 Department of Microbiology and Immunology, University of Michigan, Ann Arbor, Michigan, United States

Received  December 2002 Revised  June 2003 Published  November 2003

In this paper a mathematical model is presented that describes growth, immune escape, and siRNA treatment of tumors. The model consists of a system of nonlinear, ordinary differential equations describing tumor cells and immune effectors, as well as the immuno-stimulatory and suppressive cytokines IL-2 and TGF-$\beta$. TGF-$\beta$ suppresses the immune system by inhibiting the activation of effector cells and reducing tumor antigen expression. It also stimulates tumor growth by promoting angiogenesis, explaining the inclusion of an angiogenic switch mechanism for TGF-$\beta$ activity. The model predicts that increasing the rate of TGF-$\beta$ production for reasonable values of tumor antigenicity enhances tumor growth and its ability to escape host detection. The model is then extended to include siRNA treatment which suppresses TGF-$\beta$ production by targeting the mRNA that codes for TGF-$\beta$, thereby reducing the presence and effect of TGF-$\beta$ in tumor cells. Comparison of tumor response to multiple injections of siRNA with behavior of untreated tumors demonstrates the effectiveness of this proposed treatment strategy. A second administration method, continuous infusion, is included to contrast the ideal outcome of siRNA treatment. The model's results predict conditions under which siRNA treatment can be successful in returning an aggressive, TGF-$\beta$ producing tumor to its passive, non-immune evading state.
Citation: J.C. Arciero, T.L. Jackson, D.E. Kirschner. A mathematical model of tumor-immune evasion and siRNA treatment. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 39-58. doi: 10.3934/dcdsb.2004.4.39
 [1] Marcello Delitala, Tommaso Lorenzi. Recognition and learning in a mathematical model for immune response against cancer. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 891-914. doi: 10.3934/dcdsb.2013.18.891 [2] Sophia R-J Jang, Hsiu-Chuan Wei. On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3261-3295. doi: 10.3934/dcdsb.2021184 [3] Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 [4] Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences & Engineering, 2015, 12 (1) : 185-208. doi: 10.3934/mbe.2015.12.185 [5] Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157 [6] Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi. Mathematical modeling on helper T cells in a tumor immune system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 55-72. doi: 10.3934/dcdsb.2014.19.55 [7] Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863 [8] Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022 [9] Fabrizio Clarelli, Roberto Natalini. A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions. Mathematical Biosciences & Engineering, 2010, 7 (2) : 277-300. doi: 10.3934/mbe.2010.7.277 [10] Seema Nanda, Lisette dePillis, Ami Radunskaya. B cell chronic lymphocytic leukemia - A model with immune response. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1053-1076. doi: 10.3934/dcdsb.2013.18.1053 [11] Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences & Engineering, 2013, 10 (1) : 37-57. doi: 10.3934/mbe.2013.10.37 [12] Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925 [13] Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140 [14] Jianjun Paul Tian, Kendall Stone, Thomas John Wallin. A simplified mathematical model of solid tumor regrowth with therapies. Conference Publications, 2009, 2009 (Special) : 771-779. doi: 10.3934/proc.2009.2009.771 [15] Kaifa Wang, Yu Jin, Aijun Fan. The effect of immune responses in viral infections: A mathematical model view. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3379-3396. doi: 10.3934/dcdsb.2014.19.3379 [16] Joseph R. Zipkin, Martin B. Short, Andrea L. Bertozzi. Cops on the dots in a mathematical model of urban crime and police response. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1479-1506. doi: 10.3934/dcdsb.2014.19.1479 [17] T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 187-201. doi: 10.3934/dcdsb.2004.4.187 [18] J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263 [19] Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173 [20] Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040

2021 Impact Factor: 1.497