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2015, 12(5): 1037-1053. doi: 10.3934/mbe.2015.12.1037

## Analysis of a cancer dormancy model and control of immuno-therapy

 1 Department of Mathematics, Iowa State University, Ames, IA 50011, United States, United States

Received  July 2014 Revised  February 2015 Published  June 2015

The goal of this paper is to analyze a model of cancer-immune system interactions from [16], and to show how the introduction of control in this model can dramatically improve the hypothetical patient response and in effect prevent the cancer from growing. We examine all the equilibrium points of the model and classify them according to their stability properties. We identify an equilibrium point corresponding to a survivable amount of cancer cells which are exactly balanced by the immune response. This situation corresponds to cancer dormancy. By using Lyapunov stability theory we estimate the region of attraction of this equilibrium and propose two control laws which are able to stabilize the system effectively, improving the results of [16]. Ultimately, the analysis presented in this paper reveals that a slower, continuous introduction of antibodies over a short time scale, as opposed to mere inoculation, may lead to more efficient and safer treatments.
Citation: Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037
##### References:
 [1] A. M. Baker, et. al., Lysyl Oxidase Plays a Critical Role in Endothelial Cell Stimulation to Drive Tumor Angiogenesis,, Cancer Research, 73 (2013), 583. doi: 10.1158/0008-5472.CAN-12-2447. [2] A. D'Onofrio, A general framework for modeling tumor-immune system competition and immuno-therapy, mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032. [3] G. P. Dunn, L. J. Old and R. D. Schreiber, The three E's of cancer immuno-editing,, Annu. Rev. Immunol., 22 (2004), 329. [4] R. Eftimie, J. L. Bramson and D. J. Earn, Interactions between the immune systems and cancer: A brief review of non-spatial mathematical models,, Bull. Math. Biol., 73 (2011), 2. doi: 10.1007/s11538-010-9526-3. [5] J. Erler, et. al., Lysyl oxidase is essential for hypoxia-induced metastasis,, Nature, 440 (2006), 1222. doi: 10.1038/nature04695. [6] R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals,, IEEE Transactions on Automatic Control, 30 (1985), 747. doi: 10.1109/TAC.1985.1104057. [7] W. Hahn, Stability of Motion,, Springer Verlag, (1967). [8] T. Kailath, Linear Systems,, Prentice-Hall Information and System Sciences Series. Prentice-Hall, (1980). [9] M. Krstić, I. Kanellakopoulos and P. V. Kokotović, Nonlinear and Adaptive Control Design,, John Wiley and Sons, (1995). [10] V. A. Kuznetsov, Mathematical modeling of the development of dormant tumors and immune stimulation of their growth,, Cybern. syst. Anal, 23 (1988), 556. [11] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol, 56 (1994), 295. [12] U. Ledzewicz, M. Faraji and H. Schaettler, Mathematical model of tumor-immune interactions under chemotherapy with immune boost,, Discrete and Continuous Dynamical Systems, 18 (2013), 1031. doi: 10.3934/dcdsb.2013.18.1031. [13] U. Ledzewicz, M. Naghneian and H. Schaettler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics,, Journal of Mathematical Biology, 64 (2012), 557. doi: 10.1007/s00285-011-0424-6. [14] L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy,, J. of the National Cancer Institute, 58 (1977), 1735. [15] L. Norton, A Goempertzian model of human breast cancer growth,, Cancer Research, 48 (1988), 7067. [16] K. Page and J. Uhr, Mathematical models of cancer dormancy,, Leukemia and Lymphoma, 46 (2005), 313. doi: 10.1080/10428190400011625. [17] G. Phan et. al., Cancer regression and autoimmunity induced by cytotoxic T lymphocyte-associated antigen 4 blockade in patients with metastatic melanoma,, PNAS, 100 (2003), 8372. [18] S. Ratschan and Z. She, Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,, SIAM J. Control Optim., 48 (2010), 4377. doi: 10.1137/090749955. [19] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness,, Prentice-Hall, (1989). [20] A. Scott, J. Wolchock and L. Old, Antibody therapy of cancer,, Nature Reviews Cancer, 12 (2012), 278. doi: 10.1038/nrc3236. [21] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumor,, Biophysics, 24 (1980), 917. [22] T. Takayanagi, H. Kawamura and A. Ohuchi, Cellular automaton model of a tumor tissue consisting of tumor cells, cytoxic T lymphocytes (CTLs), and cytokine produced by CTLs,, IPSJ Trans Math Model Appl., 47 (2006), 61. [23] A. Vannelli and M. Vidyasagar, Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems,, Automatica, 21 (1985), 69. doi: 10.1016/0005-1098(85)90099-8. [24] K. P. Wilkie, A Review of Mathematical Models of Cancer-Immune Interactions in the Context of Tumor Dormancy,, Systems Biology of Tumor Dormancy, (2013). [25] V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems,, Israel Jerusalem Academic Press, (1962). [26] V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, the Netherlands, (1964). [27] , Sydney International Workshop on Math Models of Tumor-Immune System Dynamics,, January 7-10, (2013), 7.

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##### References:
 [1] A. M. Baker, et. al., Lysyl Oxidase Plays a Critical Role in Endothelial Cell Stimulation to Drive Tumor Angiogenesis,, Cancer Research, 73 (2013), 583. doi: 10.1158/0008-5472.CAN-12-2447. [2] A. D'Onofrio, A general framework for modeling tumor-immune system competition and immuno-therapy, mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032. [3] G. P. Dunn, L. J. Old and R. D. Schreiber, The three E's of cancer immuno-editing,, Annu. Rev. Immunol., 22 (2004), 329. [4] R. Eftimie, J. L. Bramson and D. J. Earn, Interactions between the immune systems and cancer: A brief review of non-spatial mathematical models,, Bull. Math. Biol., 73 (2011), 2. doi: 10.1007/s11538-010-9526-3. [5] J. Erler, et. al., Lysyl oxidase is essential for hypoxia-induced metastasis,, Nature, 440 (2006), 1222. doi: 10.1038/nature04695. [6] R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals,, IEEE Transactions on Automatic Control, 30 (1985), 747. doi: 10.1109/TAC.1985.1104057. [7] W. Hahn, Stability of Motion,, Springer Verlag, (1967). [8] T. Kailath, Linear Systems,, Prentice-Hall Information and System Sciences Series. Prentice-Hall, (1980). [9] M. Krstić, I. Kanellakopoulos and P. V. Kokotović, Nonlinear and Adaptive Control Design,, John Wiley and Sons, (1995). [10] V. A. Kuznetsov, Mathematical modeling of the development of dormant tumors and immune stimulation of their growth,, Cybern. syst. Anal, 23 (1988), 556. [11] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol, 56 (1994), 295. [12] U. Ledzewicz, M. Faraji and H. Schaettler, Mathematical model of tumor-immune interactions under chemotherapy with immune boost,, Discrete and Continuous Dynamical Systems, 18 (2013), 1031. doi: 10.3934/dcdsb.2013.18.1031. [13] U. Ledzewicz, M. Naghneian and H. Schaettler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics,, Journal of Mathematical Biology, 64 (2012), 557. doi: 10.1007/s00285-011-0424-6. [14] L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy,, J. of the National Cancer Institute, 58 (1977), 1735. [15] L. Norton, A Goempertzian model of human breast cancer growth,, Cancer Research, 48 (1988), 7067. [16] K. Page and J. Uhr, Mathematical models of cancer dormancy,, Leukemia and Lymphoma, 46 (2005), 313. doi: 10.1080/10428190400011625. [17] G. Phan et. al., Cancer regression and autoimmunity induced by cytotoxic T lymphocyte-associated antigen 4 blockade in patients with metastatic melanoma,, PNAS, 100 (2003), 8372. [18] S. Ratschan and Z. She, Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,, SIAM J. Control Optim., 48 (2010), 4377. doi: 10.1137/090749955. [19] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness,, Prentice-Hall, (1989). [20] A. Scott, J. Wolchock and L. Old, Antibody therapy of cancer,, Nature Reviews Cancer, 12 (2012), 278. doi: 10.1038/nrc3236. [21] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumor,, Biophysics, 24 (1980), 917. [22] T. Takayanagi, H. Kawamura and A. Ohuchi, Cellular automaton model of a tumor tissue consisting of tumor cells, cytoxic T lymphocytes (CTLs), and cytokine produced by CTLs,, IPSJ Trans Math Model Appl., 47 (2006), 61. [23] A. Vannelli and M. Vidyasagar, Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems,, Automatica, 21 (1985), 69. doi: 10.1016/0005-1098(85)90099-8. [24] K. P. Wilkie, A Review of Mathematical Models of Cancer-Immune Interactions in the Context of Tumor Dormancy,, Systems Biology of Tumor Dormancy, (2013). [25] V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems,, Israel Jerusalem Academic Press, (1962). [26] V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, the Netherlands, (1964). [27] , Sydney International Workshop on Math Models of Tumor-Immune System Dynamics,, January 7-10, (2013), 7.
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