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Optimal controls for a mathematical model of tumorimmune interactions under targeted chemotherapy with immune boost
B cell chronic lymphocytic leukemia  A model with immune response
1.  Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore 560065, India 
2.  Department of Mathematics, Harvey Mudd College, Claremont, CA 91711 
3.  Department of Mathematics, Pomona College, Claremont, CA, 91711, United States 
References:
[1] 
M. J. Keating, N. Chiorazzi, B. Messmer, R. N. Damle, S. L. Allen, K. R. Rai, et al., Biology and treatment of chronic lymphocytic leukemia, American Society of Hematology, [Review], 153 (2003), 153175. 
[2] 
N. Chiorazzi, K. R. Rai and M. Ferrarini, Mechanisms of disease: Chronic lymphocytic leukemia, The New England Journal of Medicine, 352 (2005), 804815, [http://www.nejm.org]. 
[3] 
F. CaligarisCappio and R. D. Favera (Eds), "Chronic Lymphocytic Leukemia," No. 294 in Current Topics in Microbiology and Immunology, Springer, 1 edition 2005. 
[4] 
C. Imai, S. Iwamoto and D. Campana, Genetic modification of primary natural killer cells overcomes inhibitory signals and induces specific killing of leukemic cells, Blood, 106 (2005), 376383. 
[5] 
S. Y. Zimmermann, R. Esser, E. Rohrbach, T. Klingebiel and U. Koehl, A novel fourcolour flow cytometric assay to determine natural killer cell or Tcellmediated cellular cytotoxicity against leukaemic cells in peripheral or bone marrow specimens containing greater than 20 Journal of Immunological Methods, 296 (2005), 6376. 
[6] 
H. Guven, M. Gilljam, B. Chambers, H. Ljunggren, B. Christensson, E. Kimby, et al., Expansion of natural killer (NK) and natural killerlike T (NKT)cell populations derived from patients with Bchronic lymphocytic leukemia (BCLL), a potential source for cellular immunotherapy, Leukemia, 17 (2003), 19731980. 
[7] 
R. J. Prestwich, F. Errington, P. Hatfieldy, A. E. Merrick, E. J. Ilett, P. J. Selby, et al., The immune system  Is it relevant to cancer development, progression and treatment? Clinical Oncology, 20 (2008), 101112. 
[8] 
E. Gitelson, C. Hammond, J. Mena, M. Lorenzo, R. Buckstein, N. L. Berinstein, et al., Chronic lymphocytic leukemiareactive T cells during disease progression and after autologous tumor cell vaccines, Clinical Cancer Research, 9 (2003), 16561665. 
[9] 
M. C. Mackey, C. Ou, L. PujoMenjouet and J. Wu, Periodic oscillations of blood cell populations in chronic myelogenous leukemia, SIAM J. Math. Anal., 38 (2006), 166187. doi: 10.1137/04061578X. 
[10] 
C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis  I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117146. doi: 10.1016/j.jtbi.2005.03.033. 
[11] 
H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction, Journal of Theoretical Biology, 227 (2004), 513523. doi: 10.1016/j.jtbi.2003.11.024. 
[12] 
C. Haurie, D. C. Dale and M. C. Mackey, Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models, Blood, 92 (1998), 26292640. 
[13] 
B. Vitale, M. Martinis, M. Antica, B. Kusic, S. Rabatic, A. Gagro, et al., Prolegomenon for chronic lymphocytic leukaemia, Scandinavian Journal of Immunology, 58 (2003), 588600. 
[14] 
M. Martinis, B. Vitale, V. Zlatic, B. Dobrosevic and K. Dodig, Mathematical model of Bcell chronic lymphocytic leukemia (CLL), Periodicum Biologorum, 107 (2005), 445450. 
[15] 
S. I. Niculescu, P. S. Kim, K. Gu, P. P. Lee and D. Levy, Stability crossing boundaries of delaty sustems modeling immune dynamics in leukemia, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 129156. doi: 10.3934/dcdsb.2010.13.129. 
[16] 
B. T. Messmer, D. Messmer, S. L. Allen, J. E. Kolitz, P. Kudalkar, D. Cesar, et al., In vivo measurements document the dynamic cellular kinetics of chronic lymphocytic leukemia B cells, The Journal of Clinical Investigation, 115 (2005), 755764. 
[17] 
H. Mellstedt and A. Choudhur, T and B cells in in Bchronic lymphocytic leukaemia: Faust, mephistopheles and the pact with the devil, Cancer Immunology, 55 (2006), 210220. 
[18] 
A. L. Shaffer III, R. M. Young and L. M. Staudt, Pathogenesis of human B cell lymphomas, Annual Review of Immunology, 30 (2012), 565610. 
[19] 
L. G. dePillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth, Cancer Research, 65 (2005), 79507958. 
[20] 
C. A. Janeway, P. Travers, M. Walport andM. J. Shlomchik, "Immunobiology," Garland Science, 6th edition 2005. 
[21] 
V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295321. 
[22] 
S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review, 62 (1994), 229243. 
[23] 
M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Papageorgopoulos, E. Wieder, et al., Directly measured kinetics of circulating T lymphocytes in normal and HIV1infected humans, Nature Medicine, 5 (1999), 8389. 
[24] 
A. M. Jamieson, P. Isnard, J. R. Dorfman, M. C. Coles and D. H. Raulet, Turnover and proliferation of NK cells in steady state and lymphopenic conditions, The Journal of Immunology, 172 (2004), 864870. 
[25] 
R. J. DeBoer, H. Mohri, D. D. Ho and A. S. Perelson, Turnover rates of B cells, T cells, and NK cells in simian immunodeficiency virusinfected and uninfected rhesus macaques, The Journal of Immunology, 170 (2003), 24792487. 
[26] 
A. K. Abbas, A. H. Lichtman and S. Pillai, "Cellular and Molecular Immunology," Saunders, 7 edition 2007. 
[27] 
A. Shimoni, H. Marcus, A. Canaan, D. Ergas, M. David, A. Berrebi, et al., A model for human Bchronic lymphocytic leukemia in human/mouse radiation chimera: Evidence for tumormediated suppression of antibody production in lowstage disease, Blood, 89 (1997), 22102218. 
[28] 
Y. Jiang, H. Shang, Z. Zhang, Y. Diao, D. Dai, W. Geng, et al., Alterations of natural killer cell and Tlymphocyte counts in adults infected with human immunodeficiency virus through blood and plasma sold in the past in China and in whom infection has progressed slowly over a long period, Clinical and Diagnostic Laboratory Immunology, 12 (2005), 12751279. 
[29] 
E. Kimby, H. Mellstedt, B. Nilsson, M. Björkholm and G. Holm, Differences in blood T and NK cell populations between chronic lymphocytic leukemia of B cell type (BCLL) and monoclonal BLymphocytes of Undetermined Significance (BMLUS), Leukemia, 3 (1989), 501504. 
[30] 
M. Hernberg, T. Muhonen, J. P. Turunen, M .HahkaKemppinen and S. Pyrhonen, The CD4+/CD8+ ratio as a prognostic factor in patients with metastatic melanoma receiving chemoimmunotherapy, Journal Of Clinical Oncology, 14 (1996), 16981696. 
[31] 
I. Tinhofer, I. Marschitz, M. Kos, T. Henn, A. Egle, A. Villunger and R. Greil, Differential sensitivity of CD41 and CD81 T lymphocytes to the killing efficacy of fas (Apo1/CD95) Ligand1 tumor cells in B chronic lymphocytic leukemia, Blood, 91 (1998), 42734281. 
[32] 
G. B. West, J. H. Brown and B. J. Enquist, A general model for the origin of allometric scaling laws in biology, Science, 276 (1997), 122126. 
[33] 
A. McClean and C. Michie, In vivo estimates of division and death rates of general human T lymphocytes, Proc. Natl. Acad. Sci. USA, 92 (1995), 37073711. 
show all references
References:
[1] 
M. J. Keating, N. Chiorazzi, B. Messmer, R. N. Damle, S. L. Allen, K. R. Rai, et al., Biology and treatment of chronic lymphocytic leukemia, American Society of Hematology, [Review], 153 (2003), 153175. 
[2] 
N. Chiorazzi, K. R. Rai and M. Ferrarini, Mechanisms of disease: Chronic lymphocytic leukemia, The New England Journal of Medicine, 352 (2005), 804815, [http://www.nejm.org]. 
[3] 
F. CaligarisCappio and R. D. Favera (Eds), "Chronic Lymphocytic Leukemia," No. 294 in Current Topics in Microbiology and Immunology, Springer, 1 edition 2005. 
[4] 
C. Imai, S. Iwamoto and D. Campana, Genetic modification of primary natural killer cells overcomes inhibitory signals and induces specific killing of leukemic cells, Blood, 106 (2005), 376383. 
[5] 
S. Y. Zimmermann, R. Esser, E. Rohrbach, T. Klingebiel and U. Koehl, A novel fourcolour flow cytometric assay to determine natural killer cell or Tcellmediated cellular cytotoxicity against leukaemic cells in peripheral or bone marrow specimens containing greater than 20 Journal of Immunological Methods, 296 (2005), 6376. 
[6] 
H. Guven, M. Gilljam, B. Chambers, H. Ljunggren, B. Christensson, E. Kimby, et al., Expansion of natural killer (NK) and natural killerlike T (NKT)cell populations derived from patients with Bchronic lymphocytic leukemia (BCLL), a potential source for cellular immunotherapy, Leukemia, 17 (2003), 19731980. 
[7] 
R. J. Prestwich, F. Errington, P. Hatfieldy, A. E. Merrick, E. J. Ilett, P. J. Selby, et al., The immune system  Is it relevant to cancer development, progression and treatment? Clinical Oncology, 20 (2008), 101112. 
[8] 
E. Gitelson, C. Hammond, J. Mena, M. Lorenzo, R. Buckstein, N. L. Berinstein, et al., Chronic lymphocytic leukemiareactive T cells during disease progression and after autologous tumor cell vaccines, Clinical Cancer Research, 9 (2003), 16561665. 
[9] 
M. C. Mackey, C. Ou, L. PujoMenjouet and J. Wu, Periodic oscillations of blood cell populations in chronic myelogenous leukemia, SIAM J. Math. Anal., 38 (2006), 166187. doi: 10.1137/04061578X. 
[10] 
C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis  I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117146. doi: 10.1016/j.jtbi.2005.03.033. 
[11] 
H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction, Journal of Theoretical Biology, 227 (2004), 513523. doi: 10.1016/j.jtbi.2003.11.024. 
[12] 
C. Haurie, D. C. Dale and M. C. Mackey, Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models, Blood, 92 (1998), 26292640. 
[13] 
B. Vitale, M. Martinis, M. Antica, B. Kusic, S. Rabatic, A. Gagro, et al., Prolegomenon for chronic lymphocytic leukaemia, Scandinavian Journal of Immunology, 58 (2003), 588600. 
[14] 
M. Martinis, B. Vitale, V. Zlatic, B. Dobrosevic and K. Dodig, Mathematical model of Bcell chronic lymphocytic leukemia (CLL), Periodicum Biologorum, 107 (2005), 445450. 
[15] 
S. I. Niculescu, P. S. Kim, K. Gu, P. P. Lee and D. Levy, Stability crossing boundaries of delaty sustems modeling immune dynamics in leukemia, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 129156. doi: 10.3934/dcdsb.2010.13.129. 
[16] 
B. T. Messmer, D. Messmer, S. L. Allen, J. E. Kolitz, P. Kudalkar, D. Cesar, et al., In vivo measurements document the dynamic cellular kinetics of chronic lymphocytic leukemia B cells, The Journal of Clinical Investigation, 115 (2005), 755764. 
[17] 
H. Mellstedt and A. Choudhur, T and B cells in in Bchronic lymphocytic leukaemia: Faust, mephistopheles and the pact with the devil, Cancer Immunology, 55 (2006), 210220. 
[18] 
A. L. Shaffer III, R. M. Young and L. M. Staudt, Pathogenesis of human B cell lymphomas, Annual Review of Immunology, 30 (2012), 565610. 
[19] 
L. G. dePillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth, Cancer Research, 65 (2005), 79507958. 
[20] 
C. A. Janeway, P. Travers, M. Walport andM. J. Shlomchik, "Immunobiology," Garland Science, 6th edition 2005. 
[21] 
V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295321. 
[22] 
S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review, 62 (1994), 229243. 
[23] 
M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Papageorgopoulos, E. Wieder, et al., Directly measured kinetics of circulating T lymphocytes in normal and HIV1infected humans, Nature Medicine, 5 (1999), 8389. 
[24] 
A. M. Jamieson, P. Isnard, J. R. Dorfman, M. C. Coles and D. H. Raulet, Turnover and proliferation of NK cells in steady state and lymphopenic conditions, The Journal of Immunology, 172 (2004), 864870. 
[25] 
R. J. DeBoer, H. Mohri, D. D. Ho and A. S. Perelson, Turnover rates of B cells, T cells, and NK cells in simian immunodeficiency virusinfected and uninfected rhesus macaques, The Journal of Immunology, 170 (2003), 24792487. 
[26] 
A. K. Abbas, A. H. Lichtman and S. Pillai, "Cellular and Molecular Immunology," Saunders, 7 edition 2007. 
[27] 
A. Shimoni, H. Marcus, A. Canaan, D. Ergas, M. David, A. Berrebi, et al., A model for human Bchronic lymphocytic leukemia in human/mouse radiation chimera: Evidence for tumormediated suppression of antibody production in lowstage disease, Blood, 89 (1997), 22102218. 
[28] 
Y. Jiang, H. Shang, Z. Zhang, Y. Diao, D. Dai, W. Geng, et al., Alterations of natural killer cell and Tlymphocyte counts in adults infected with human immunodeficiency virus through blood and plasma sold in the past in China and in whom infection has progressed slowly over a long period, Clinical and Diagnostic Laboratory Immunology, 12 (2005), 12751279. 
[29] 
E. Kimby, H. Mellstedt, B. Nilsson, M. Björkholm and G. Holm, Differences in blood T and NK cell populations between chronic lymphocytic leukemia of B cell type (BCLL) and monoclonal BLymphocytes of Undetermined Significance (BMLUS), Leukemia, 3 (1989), 501504. 
[30] 
M. Hernberg, T. Muhonen, J. P. Turunen, M .HahkaKemppinen and S. Pyrhonen, The CD4+/CD8+ ratio as a prognostic factor in patients with metastatic melanoma receiving chemoimmunotherapy, Journal Of Clinical Oncology, 14 (1996), 16981696. 
[31] 
I. Tinhofer, I. Marschitz, M. Kos, T. Henn, A. Egle, A. Villunger and R. Greil, Differential sensitivity of CD41 and CD81 T lymphocytes to the killing efficacy of fas (Apo1/CD95) Ligand1 tumor cells in B chronic lymphocytic leukemia, Blood, 91 (1998), 42734281. 
[32] 
G. B. West, J. H. Brown and B. J. Enquist, A general model for the origin of allometric scaling laws in biology, Science, 276 (1997), 122126. 
[33] 
A. McClean and C. Michie, In vivo estimates of division and death rates of general human T lymphocytes, Proc. Natl. Acad. Sci. USA, 92 (1995), 37073711. 
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