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Analysis of a cancer dormancy model and control of immunotherapy
1.  Department of Mathematics, Iowa State University, Ames, IA 50011, United States, United States 
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: 10.1158/00085472.CAN122447. 
[2] 
A. D'Onofrio, A general framework for modeling tumorimmune system competition and immunotherapy, 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 immunoediting,, 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 nonspatial mathematical models,, Bull. Math. Biol., 73 (2011), 2. doi: 10.1007/s1153801095263. 
[5] 
J. Erler, et. al., Lysyl oxidase is essential for hypoxiainduced 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]  
[8] 
T. Kailath, Linear Systems,, PrenticeHall Information and System Sciences Series. PrenticeHall, (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 tumorimmune 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 tumorimmune dynamics,, Journal of Mathematical Biology, 64 (2012), 557. doi: 10.1007/s0028501104246. 
[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 lymphocyteassociated 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 Lyapunovlike functions,, SIAM J. Control Optim., 48 (2010), 4377. doi: 10.1137/090749955. 
[19] 
S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness,, PrenticeHall, (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: 10.1016/00051098(85)900998. 
[24] 
K. P. Wilkie, A Review of Mathematical Models of CancerImmune 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 TumorImmune System Dynamics,, January 710, (2013), 7. 
show all references
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: 10.1158/00085472.CAN122447. 
[2] 
A. D'Onofrio, A general framework for modeling tumorimmune system competition and immunotherapy, 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 immunoediting,, 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 nonspatial mathematical models,, Bull. Math. Biol., 73 (2011), 2. doi: 10.1007/s1153801095263. 
[5] 
J. Erler, et. al., Lysyl oxidase is essential for hypoxiainduced 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]  
[8] 
T. Kailath, Linear Systems,, PrenticeHall Information and System Sciences Series. PrenticeHall, (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 tumorimmune 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 tumorimmune dynamics,, Journal of Mathematical Biology, 64 (2012), 557. doi: 10.1007/s0028501104246. 
[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 lymphocyteassociated 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 Lyapunovlike functions,, SIAM J. Control Optim., 48 (2010), 4377. doi: 10.1137/090749955. 
[19] 
S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness,, PrenticeHall, (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: 10.1016/00051098(85)900998. 
[24] 
K. P. Wilkie, A Review of Mathematical Models of CancerImmune 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 TumorImmune System Dynamics,, January 710, (2013), 7. 
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