Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T00:23:13.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolutionary Dynamics of Cancer Cell Populations under Immune Selection Pressure and Optimal Control of Chemotherapy

Published online by Cambridge University Press:  20 June 2014

G. Dimitriu ,
T. Lorenzi  and
R. Ştefănescu
G. Dimitriu*
Affiliation:
“Grigore T. Popa” University of Medicine and Pharmacy, Department of Mathematics and Informatics, Iaşi 700115, Romania
T. Lorenzi
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France INRIA-Paris-Rocquencourt, EPC MAMBA, Domaine de Voluceau, BP105, 78153 Le Chesnay Cedex
R. Ştefănescu
Affiliation:
Virginia Tech, Department of Computer Science, Blacksburg, VA, 24060, USA
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

Increasing experimental evidence suggests that epigenetic and microenvironmental factors play a key role in cancer progression. In this respect, it is now generally recognized that the immune system can act as an additional selective pressure, which modulates tumor development and leads, through cancer immunoediting, to the selection for resistance to immune effector mechanisms. This may have serious implications for the design of effective anti-cancer protocols. Motivated by these considerations, we present a mathematical model for the dynamics of cancer and immune cells under the effects of chemotherapy and immunity-boosters. Tumor cells are modeled as a population structured by a continuous phenotypic trait, that is related to the level of resistance to receptor-induced cell death triggered by effector lymphocytes. The level of resistance can vary over time due to the effects of epigenetic modifications. In the asymptotic regime of small epimutations, we highlight the ability of the model to reproduce cancer immunoediting. In an optimal control framework, we tackle the problem of designing effective anti-cancer protocols. The results obtained suggest that chemotherapeutic drugs characterized by high cytotoxic effects can be useful for treating tumors of large size. On the other hand, less cytotoxic chemotherapy in combination with immunity-boosters can be effective against tumors of smaller size. Taken together, these results support the development of therapeutic protocols relying on combinations of less cytotoxic agents and immune-boosters to fight cancer in the early stages.

Keywords

chemotherapyimmunity-boosterscancer immunoeditingcancer modelingoptimal control
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 4: Optimal control , 2014 , pp. 88 - 104
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, B.M., Banks, H.T., Kwon, H.-D., Tran, H.T.. Dynamic multidrug therapies for HIV: optimal and STI control approaches. Math. Biosci. Eng., 1 (2004), 223–241. Google ScholarPubMed
Antipov, A.V., Bratus, A.S.. Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor. Zh. Vychisl. Mat. Mat. Fiz., 49 (2009), 1907–1919. Google Scholar
Billy, F., Clairambault, J.. Designing proliferating cell population models with functional targets for control by anti-cancer drugs. Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 865–889. CrossRefGoogle Scholar
Burden, T., Ernstberger, J., Fister, K.R.. Optimal control applied to immunotherapy. Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 135–146. Google Scholar
Castiglione, F., Piccoli, B.. Optimal control in a model of dendritic cell transfection cancer immunotherapy. Bull. Math. Biol., 68 (2006), 255–274. CrossRefGoogle Scholar
Coldman, A.J., Murray, J.M.. Optimal control for a stochastic model of cancer chemotherapy. Math. Biosci., 168 (2000), 187–200. CrossRefGoogle ScholarPubMed
Costa, M., Boldrini, J., Bassanezi, R.. Drug kinetics and drug resistance in optimal chemotherapy. Math. Biosci., 125 (1995), 191–209. CrossRefGoogle Scholar
Delitala, M., Lorenzi, T.. A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions. J. Theoret. Biol., 297 (2012), 88-102. CrossRefGoogle ScholarPubMed
Desvillettes, L., Jabin, P.E., Mischler, S., Raoul, G.. On selection dynamics for continuous structured populations. Commun. Math. Sci., 6 (2008), 729–747. CrossRefGoogle Scholar
Diekmann, O., Jabin, P.E., Mischler, S., Perthame, B.. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. Theor. Pop. Biol., 67 (2005), 257–271. CrossRefGoogle Scholar
Dimitriu, G.. Numerical approximation of the optimal inputs for an identification problem. Intern. J. Computer Math., 70 (1998), 197–209. CrossRefGoogle Scholar
Dua, P., Dua, V., Pistikopoulos, E.. Optimal delivery of chemotherapeutic agents in cancer. Comput. Chem. Eng., 32 (2008), 99–107. CrossRefGoogle Scholar
Dunn, G.P., Bruce, A.T., Ikeda, H., Old, L.J., Schreiber, R.D.. Cancer immunoediting: from immunosurveillance to tumor escape. Nature Immunol., 3 (2002), 991–998. CrossRefGoogle ScholarPubMed
DuPage, M., Mazumdar, C., Schmidt, L.M., Cheung, A.F., Jacks, T.. Expression of tumour-specific antigens underlies cancer immunoediting. Nature, 482 (2012), 405–9. CrossRefGoogle ScholarPubMed
Engelhart, M., Lebiedz, D., Sager, S.. Optimal control for selected cancer chemotherapy ODE models: A view on the potential of optimal schedules and choice of objective function. Math. Biosci., 229 (2011), 123–134. CrossRefGoogle ScholarPubMed
Fister, K.R., Donnelly, J.. Immunotherapy: an optimal control theory approach. Math. Biosci. Eng., 2 (2005), 499–510. Google ScholarPubMed
Fister, K.R., Panetta, J.C.. Optimal control applied to cell-cycle-specific cancer chemotherapy. SIAM J. Appl. Math., 60 (2000), 1059–1072. Google Scholar
Fister, K.R., Panetta, J.C.. Optimal control applied to competing chemotherapeutic cell-kill strategies. SIAM J. Appl. Math., 63 (2003), 1954–1971. Google Scholar
W.H. Fleming, R.W. Rishel. Deterministic and Stochastic Optimal Control. Springer-Verlag, 1975.
Ghaffari, A., Naserifar, N.. Optimal therapeutic protocols in cancer immunotherapy. Comput. Biol. Med., 40 (2010), 261–270. CrossRefGoogle ScholarPubMed
J. Goldie, A. Coldman. Drug resistance in cancer: mechanisms and models. Cambridge University Press, 1998.
Gottesman, M.. Mechanisms of cancer drug resistance. Annu. Rev. Med., 53 (2002), 615–627. CrossRefGoogle ScholarPubMed
Igney, F.H., Krammer, P.H.. Immune escape of tumors: apoptosis resistance and tumor counterattack. J. Leukoc. Biol., 71 (2002), 907–20. Google Scholar
L.S. Jennings, M.E. Fisher, K.L. Teo, C.J. Goh. MISER3 Optimal Control Software: Theory and User Manual. Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia, 2004.
M.I. Kamien, N.L. Schwartz. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Advanced Textbooks in Economics. Second ed., vol. 31. North-Holland, 1991.
Komarova, N., Wodarz, D.. Drug resistance in cancer: Principles of emergence and prevention. Proc Natl Acad Sci USA, 102 (2005), 9714–9719. CrossRefGoogle Scholar
Ledzewicz, U., d’Onofrio, A., Maurer, H., Schäettler, H.. On optimal delivery of combination therapy for tumors. Math. Biosci., 222 (2009), 13–26. Google Scholar
Ledzewicz, U., Naghnaeian, M., Schättler, H.. An optimal control approach to cancer treatment under immunological activity. Appl. Math., 38 (2011), 17–31. Google Scholar
T. Lorenzi, A. Lorz, G. Restori. Asymptotic dynamics in populations structured by sensitivity to global warming and habitat shrinking. Acta Appl. Math., 2013, DOI 10.1007/s10440-013-9849-9.
Liu, K.. Role of apoptosis resistance in immune evasion and metastasis of colorectal cancer. World J. Gastrointest. Oncol., 15 (2010), 399–406. CrossRefGoogle Scholar
Lorz, A., Lorenzi, T., Hochberg, M.E., Clairambault, J., Perthame, B.. Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377–399. CrossRefGoogle Scholar
A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame. Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. preprint, 2014.
D.L. Lukes. Differential Equations: Classical to Controlled, vol. 162. Academic Press, 1982.
R. Martin, K. Teo. Optimal Drug Administration in Cancer Chemotherapy. World Scientific, Singapore, 1994.
Matveev, A., Savkin, A.. Application of optimal control theory to analysis of cancer chemotherapy regimens. Syst. Control Lett., 46 (2002), 311–321. CrossRefGoogle Scholar
Merlo, L., Pepper, J., Reid, B., Maley, C.. Cancer as an evolutionary and ecological process. Nat. Rev Cancer, 6 (2006), 924–935. CrossRefGoogle ScholarPubMed
Murray, J.. Optimal control for a cancer chemotherapy problem with general growth and loss functions. Math. Biosci., 98 (1990), 273–287. CrossRefGoogle Scholar
de Pillis, L.G., Gu, W., Fister, K.R., Head, T., Maples, K., Murugan, A., Neal, T., Yoshida, K.. Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls. Math. Biosci., 209 (2007), 292–315. CrossRefGoogle Scholar
de Pillis, L.G., Gu, W., Radunskaya, A.E.. Mixed immunotherapy and chemotherapy of tumors: modelling, applications and biological interpretations. J. Theor. Biol., 238 (2006), 841–862. CrossRefGoogle Scholar
de Pillis, L.G., Radunskaya, A.E., Wiseman, C.L.. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res., 65 (2005), 7950–7958. CrossRefGoogle Scholar
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes. Gordon and Breach, 1962.
Shankaran, V., Ikeda, H., Bruce, A.T., White, J.M., Swanson, P.E., Old, L.J., Schreiber, R.D.. IFNgamma and lymphocytes prevent primary tumour development and shape tumour immunogenicity, Nature 410 (2001), 1107–1111. CrossRefGoogle ScholarPubMed
Swan, G.W.. Role of optimal control theory in cancer chemotherapy. Math. Biosci., 101 (1990), 237–284. CrossRefGoogle ScholarPubMed
Szymanska, Z.. Analysis of immunotherapy models in the context of cancer dynamics. Int. J. Appl. Math. Comput. Sci., 13 (2003), 407–418. Google Scholar
Zindl, C.L., Chaplin, D.D.. Tumor immune evasion. Science, 328 (2010), 697–698. CrossRefGoogle Scholar