June  2013, 18(4): 891-914. doi: 10.3934/dcdsb.2013.18.891

Recognition and learning in a mathematical model for immune response against cancer

1. 

Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, Italy

Received  August 2012 Revised  September 2012 Published  February 2013

This paper presents a mathematical model for immune response against cancer aimed at reproducing emerging phenomena arising from the interactions between tumor and immune cells. The model is stated in terms of integro-differential equations and describes the dynamics of tumor cells, characterized by heterogeneous antigenic expressions, antigen-presenting cells and T-cells. Asymptotic analysis and simulations, developed with an exploratory aim, are addressed to verify the consistency of the model outputs as well as to provide biological insights into the mechanisms that rule tumor-immune interactions. In particular, the present model seems able to mimic the recognition, learning and memory aspects of immune response and highlights how the immune system might act as an additional selective pressure leading, eventually, to the selection for the most resistant cancer clones.
Citation: 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
References:
[1]

E. Agliari, A. Barra, F. Guerra and F. Moauro, A thermodynamical perspective of immune capabilities, J. Theor. Biol., 287 (2010), 48-63.

[2]

N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory to modeling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008), 183-206.

[3]

A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach,'' Birkhäuser, Boston, 2006.

[4]

C. Bianca and M. Delitala, On the modelling genetic mutations and immune system competition, Comput. Math. Appl., 61 (2011) 2362-2375. doi: 10.1016/j.camwa.2011.01.024.

[5]

S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076. doi: 10.1007/s11538-008-9344-z.

[6]

R. E. Callard and A. J. Yates, Immunology and mathematics: Crossing the divide, Immunology, 115 (2005), 21-33.

[7]

V. Calvez, A. Korobeinikov and P. K. Maini, Cluster formation for multi-strain infections with cross-immunity, J. Theor. Biol., 233 (2005), 75-83. doi: 10.1016/j.jtbi.2004.09.016.

[8]

C. Cattani, A. Ciancio and A. d'Onofrio, Metamodeling the learning-hiding competition between tumours and the immune system: A kinematic approach, Math. Comput. Model., 52 (2010), 62-69. doi: 10.1016/j.mcm.2010.01.012.

[9]

A. K. Chakraborty, M. L. Dustin and A. S. Shaw, In Silico models in molecular and cellular immunology: Successes, promises, and challenges, Nat. Immunol., 4 (2003), 933-936.

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A. K. Chakraborty and A. Kosmrlj, Statistical mechanical concepts in immunology, Annu. Rev. Phys. Chem., 61 (2010), 283-303.

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M. A. J. Chaplain and A. Matzavinos, Mathematical modelling of spatio-temporal phenomena in tumour immunology, Lect. Notes Math., 1872 (2006), 131-183, Springer-Verlag Berlin Heidelberg. doi: 10.1007/11561606_4.

[12]

D. Chowdhury, M. Sahimi and D. Stauffer, A discrete model for immune surveillance, tumor immunity and cancer, J. Theor. Biol., 152 (1991), 263-270.

[13]

L. G. de Pillis, D. G. Mallet and A. E. Radunskaya, Spatial tumor-immune modeling, Comput. Math. Methods Med., 7 (2006), 159-176. doi: 10.1080/10273660600968978.

[14]

M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theor. Biol., 297 (2012), 88-102. doi: 10.1016/j.jtbi.2011.11.022.

[15]

L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.

[16]

G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nat. Immunol., 3 (2002), 991-998.

[17]

P. A. W. Edwards, Heterogeneous expression of cell-surface antigens in normal epithelia and their tumours, revealed by monoclonal antibodies, Br. J. Cancer, 51 (1985), 149-160.

[18]

S. Eikenberry, C. Thalhauser and Y. Kuang, Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma, PLoS Comput. Biol., 5 (2009), e1000362. doi: 10.1371/journal.pcbi.1000362.

[19]

A. H. L. Erickson, A. Wise, S. Fleming, M. Baird, Z. Lateef, A. Molinaro, M. Teboh-Ewungkem and L. de Pillis, A preliminary mathematical model of skin dendritic cell tracking and induction of t cell immunity, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 323-336. doi: 10.3934/dcdsb.2009.12.323.

[20]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646-674.

[21]

M. Herrero, On the role of mathematics in biology, J. Math. Biol., 54 (2007), 887-889. doi: 10.1007/s00285-007-0095-5.

[22]

W. Hu, W. Zhong, F. Wang, L. Li and Y. Shao, In silico synergism and antagonism of an anti-tumour system intervened by coupling immunotherapy and chemotherapy: A mathematical modelling approach, Bull. Math. Biol., (2011). doi: 10.1007/s11538-011-9693-x.

[23]

M. Kaufman, J. Urbain and R. Thomas, Towards a logical analysis of the immune response, J. Theor. Biol., 114 (1985), 527-561. doi: 10.1016/S0022-5193(85)80042-4.

[24]

T. J. Kindt, R. A. Goldsby, B. A. Osborne and J. Kuby, "Kuby Immunology," W. H. Freeman and Company, 2005.

[25]

M. Kolev, Mathematical modeling of the competition between acquired immunity and cancer, Int. J. Appl. Math. Comput. Sci., 13 (2003), 289-296.

[26]

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), 295-321.

[27]

J. Kzhyshkowska, A. Marciniak-Czochra and A. Gratchev, Perspectives of mathematical modelling for understanding of macrophage function, Immunobiology, 212 (2007), 813-825.

[28]

D. G. Mallet and L. G. de Pillis, A cellular automata model of tumor-immune system interactions, J. Theor. Biol., 239 (2006), 334-350. doi: 10.1016/j.jtbi.2005.08.002.

[29]

D. Mason, A very high level of crossreactivity is an essential feature of the T-cell receptor, Immunology today, 19 (1998), 395-404.

[30]

A. Matzavinos, M. A.J . Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumor, Math. Med. Biol., 21 (2004), 1-34.

[31]

L. M. Merlo, J. W. Pepper, B. J. Reid and C. C. Maley, Cancer as an evolutionary and ecological process, Nat. Rev. Cancer, 6 (2006), 924-935.

[32]

R. K. Oldham and R. O. Dillman (Eds.), "Principles of Cancer Biotherapy,'' $3^{rd}$ edition, Kluwer Academic Publishers, The Netherlands, 1997.

[33]

F. Pappalardo, S. Musumeci and S. Motta, Modeling immune system control of atherogenesis, Bioinformatics, 24 (2008), 1715-1721.

[34]

A. Perelson and G. Weisbuch, Immunology for physicists, Rev. Mod. Phys., 69 (1997), 1219-1268.

[35]

B. Perthame, "Transport Equations in Biology,'' Birkhäuser, Basel, 2007.

[36]

A. Plesa , G. Ciuperca, S. Genieys, V. Louvet, L. Pujo-Menjouet, C. Dumontet and V. Volpert, Diagnostics of the AML with immunophenotypical data, Math. Mod. Nat. Phen., 2 (2006), 104-123. doi: 10.1051/mmnp:2008006.

[37]

W. R. Welch, J. M. Niloff, D. Anderson, A. Battailea, S. Emery, R. C. Knapp and R. C. Bast, Antigenic heterogeneity in human ovarian cancer, Gynecol. Oncol., 38 (1990), 12-16.

[38]

L. Wooldridge, J. Ekeruche-Makinde, H. A. van den Berg, A. Skowera, J. J. Miles, M. P. Tan, G. Dolton, M. Clement, S. Llewellyn-Lacey, D. A. Price, et al., A single autoimmune t cell receptor recognizes more than a million different peptides, Journal of Biological Chemistry, 287 (2012), 1168-1177.

show all references

References:
[1]

E. Agliari, A. Barra, F. Guerra and F. Moauro, A thermodynamical perspective of immune capabilities, J. Theor. Biol., 287 (2010), 48-63.

[2]

N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory to modeling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008), 183-206.

[3]

A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach,'' Birkhäuser, Boston, 2006.

[4]

C. Bianca and M. Delitala, On the modelling genetic mutations and immune system competition, Comput. Math. Appl., 61 (2011) 2362-2375. doi: 10.1016/j.camwa.2011.01.024.

[5]

S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076. doi: 10.1007/s11538-008-9344-z.

[6]

R. E. Callard and A. J. Yates, Immunology and mathematics: Crossing the divide, Immunology, 115 (2005), 21-33.

[7]

V. Calvez, A. Korobeinikov and P. K. Maini, Cluster formation for multi-strain infections with cross-immunity, J. Theor. Biol., 233 (2005), 75-83. doi: 10.1016/j.jtbi.2004.09.016.

[8]

C. Cattani, A. Ciancio and A. d'Onofrio, Metamodeling the learning-hiding competition between tumours and the immune system: A kinematic approach, Math. Comput. Model., 52 (2010), 62-69. doi: 10.1016/j.mcm.2010.01.012.

[9]

A. K. Chakraborty, M. L. Dustin and A. S. Shaw, In Silico models in molecular and cellular immunology: Successes, promises, and challenges, Nat. Immunol., 4 (2003), 933-936.

[10]

A. K. Chakraborty and A. Kosmrlj, Statistical mechanical concepts in immunology, Annu. Rev. Phys. Chem., 61 (2010), 283-303.

[11]

M. A. J. Chaplain and A. Matzavinos, Mathematical modelling of spatio-temporal phenomena in tumour immunology, Lect. Notes Math., 1872 (2006), 131-183, Springer-Verlag Berlin Heidelberg. doi: 10.1007/11561606_4.

[12]

D. Chowdhury, M. Sahimi and D. Stauffer, A discrete model for immune surveillance, tumor immunity and cancer, J. Theor. Biol., 152 (1991), 263-270.

[13]

L. G. de Pillis, D. G. Mallet and A. E. Radunskaya, Spatial tumor-immune modeling, Comput. Math. Methods Med., 7 (2006), 159-176. doi: 10.1080/10273660600968978.

[14]

M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theor. Biol., 297 (2012), 88-102. doi: 10.1016/j.jtbi.2011.11.022.

[15]

L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.

[16]

G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nat. Immunol., 3 (2002), 991-998.

[17]

P. A. W. Edwards, Heterogeneous expression of cell-surface antigens in normal epithelia and their tumours, revealed by monoclonal antibodies, Br. J. Cancer, 51 (1985), 149-160.

[18]

S. Eikenberry, C. Thalhauser and Y. Kuang, Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma, PLoS Comput. Biol., 5 (2009), e1000362. doi: 10.1371/journal.pcbi.1000362.

[19]

A. H. L. Erickson, A. Wise, S. Fleming, M. Baird, Z. Lateef, A. Molinaro, M. Teboh-Ewungkem and L. de Pillis, A preliminary mathematical model of skin dendritic cell tracking and induction of t cell immunity, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 323-336. doi: 10.3934/dcdsb.2009.12.323.

[20]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646-674.

[21]

M. Herrero, On the role of mathematics in biology, J. Math. Biol., 54 (2007), 887-889. doi: 10.1007/s00285-007-0095-5.

[22]

W. Hu, W. Zhong, F. Wang, L. Li and Y. Shao, In silico synergism and antagonism of an anti-tumour system intervened by coupling immunotherapy and chemotherapy: A mathematical modelling approach, Bull. Math. Biol., (2011). doi: 10.1007/s11538-011-9693-x.

[23]

M. Kaufman, J. Urbain and R. Thomas, Towards a logical analysis of the immune response, J. Theor. Biol., 114 (1985), 527-561. doi: 10.1016/S0022-5193(85)80042-4.

[24]

T. J. Kindt, R. A. Goldsby, B. A. Osborne and J. Kuby, "Kuby Immunology," W. H. Freeman and Company, 2005.

[25]

M. Kolev, Mathematical modeling of the competition between acquired immunity and cancer, Int. J. Appl. Math. Comput. Sci., 13 (2003), 289-296.

[26]

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), 295-321.

[27]

J. Kzhyshkowska, A. Marciniak-Czochra and A. Gratchev, Perspectives of mathematical modelling for understanding of macrophage function, Immunobiology, 212 (2007), 813-825.

[28]

D. G. Mallet and L. G. de Pillis, A cellular automata model of tumor-immune system interactions, J. Theor. Biol., 239 (2006), 334-350. doi: 10.1016/j.jtbi.2005.08.002.

[29]

D. Mason, A very high level of crossreactivity is an essential feature of the T-cell receptor, Immunology today, 19 (1998), 395-404.

[30]

A. Matzavinos, M. A.J . Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumor, Math. Med. Biol., 21 (2004), 1-34.

[31]

L. M. Merlo, J. W. Pepper, B. J. Reid and C. C. Maley, Cancer as an evolutionary and ecological process, Nat. Rev. Cancer, 6 (2006), 924-935.

[32]

R. K. Oldham and R. O. Dillman (Eds.), "Principles of Cancer Biotherapy,'' $3^{rd}$ edition, Kluwer Academic Publishers, The Netherlands, 1997.

[33]

F. Pappalardo, S. Musumeci and S. Motta, Modeling immune system control of atherogenesis, Bioinformatics, 24 (2008), 1715-1721.

[34]

A. Perelson and G. Weisbuch, Immunology for physicists, Rev. Mod. Phys., 69 (1997), 1219-1268.

[35]

B. Perthame, "Transport Equations in Biology,'' Birkhäuser, Basel, 2007.

[36]

A. Plesa , G. Ciuperca, S. Genieys, V. Louvet, L. Pujo-Menjouet, C. Dumontet and V. Volpert, Diagnostics of the AML with immunophenotypical data, Math. Mod. Nat. Phen., 2 (2006), 104-123. doi: 10.1051/mmnp:2008006.

[37]

W. R. Welch, J. M. Niloff, D. Anderson, A. Battailea, S. Emery, R. C. Knapp and R. C. Bast, Antigenic heterogeneity in human ovarian cancer, Gynecol. Oncol., 38 (1990), 12-16.

[38]

L. Wooldridge, J. Ekeruche-Makinde, H. A. van den Berg, A. Skowera, J. J. Miles, M. P. Tan, G. Dolton, M. Clement, S. Llewellyn-Lacey, D. A. Price, et al., A single autoimmune t cell receptor recognizes more than a million different peptides, Journal of Biological Chemistry, 287 (2012), 1168-1177.

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