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 & 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.   Google Scholar

[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.   Google Scholar

[3]

A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach,'', Birkhäuser, (2006).   Google Scholar

[4]

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

[5]

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

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

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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.  doi: 10.1016/j.mcm.2010.01.012.  Google Scholar

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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.   Google Scholar

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

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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.  doi: 10.1007/11561606_4.  Google Scholar

[12]

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

[13]

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

[14]

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

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

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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.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1371/journal.pcbi.1000362.  Google Scholar

[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.  doi: 10.3934/dcdsb.2009.12.323.  Google Scholar

[20]

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

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

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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.  Google Scholar

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

[24]

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

[25]

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

[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.   Google Scholar

[27]

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

[28]

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

[29]

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

[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.   Google Scholar

[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.   Google Scholar

[32]

R. K. Oldham and R. O. Dillman (Eds.), "Principles of Cancer Biotherapy,'', $3^{rd}$ edition, (1997).   Google Scholar

[33]

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

[34]

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

[35]

B. Perthame, "Transport Equations in Biology,'', Birkhäuser, (2007).   Google Scholar

[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.  doi: 10.1051/mmnp:2008006.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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.   Google Scholar

[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.   Google Scholar

[3]

A. Bellouquid and M. Delitala, "Modelling Complex Multicellular Systems - A Kinetic Theory Approach,'', Birkhäuser, (2006).   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1016/j.mcm.2010.01.012.  Google Scholar

[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.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1371/journal.pcbi.1000362.  Google Scholar

[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.  doi: 10.3934/dcdsb.2009.12.323.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.   Google Scholar

[27]

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

[28]

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

[29]

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

[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.   Google Scholar

[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.   Google Scholar

[32]

R. K. Oldham and R. O. Dillman (Eds.), "Principles of Cancer Biotherapy,'', $3^{rd}$ edition, (1997).   Google Scholar

[33]

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

[34]

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

[35]

B. Perthame, "Transport Equations in Biology,'', Birkhäuser, (2007).   Google Scholar

[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.  doi: 10.1051/mmnp:2008006.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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