Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-19T22:43:33.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Populational adaptive evolution, chemotherapeutic resistanceand multiple anti-cancer therapies

Published online by Cambridge University Press:  11 January 2013

Alexander Lorz ,
Tommaso Lorenzi ,
Michael E. Hochberg ,
Jean Clairambault  and
Benoît Perthame
Alexander Lorz
Affiliation:
UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu 75252 Paris Cedex 05, France.. [email protected] INRIA-Rocquencourt, EPI BANG, France.; [email protected]
Tommaso Lorenzi
Affiliation:
Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, I10129 Torino, Italy.; [email protected]
Michael E. Hochberg
Affiliation:
Institut des Sciences de l’Evolution, CNRS, Université Montpellier 2, Place Eugene Bataillon, 34095 Montpellier, France. Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, New Mexico, USA.; [email protected]
Jean Clairambault
Affiliation:
UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu 75252 Paris Cedex 05, France.. [email protected] INRIA-Rocquencourt, EPI BANG, France.; [email protected]
Benoît Perthame
Affiliation:
UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu 75252 Paris Cedex 05, France.. [email protected] INRIA-Rocquencourt, EPI BANG, France.; [email protected] Institut Universitaire de France, France. ; [email protected]
Get access

Abstract

Resistance to chemotherapies, particularly to anticancer treatments, is an increasingmedical concern. Among the many mechanisms at work in cancers, one of the most importantis the selection of tumor cells expressing resistance genes or phenotypes. Motivated bythe theory of mutation-selection in adaptive evolution, we propose a model based on acontinuous variable that represents the expression level of a resistance gene (or genes,yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects ofchemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work bydemonstrating how qualitatively different actions of chemotherapeutic and cytostatictreatments may induce different levels of resistance. The mathematical interest of ourstudy is in the formalism of constrained Hamilton–Jacobi equations in the framework ofviscosity solutions. We derive the long-term temporal dynamics of the fittest traits inthe regime of small mutations. In the context of adaptive cancer management, we alsoanalyse whether an optimal drug level is better than the maximal tolerated dose.

Keywords

Mathematical oncologyadaptive evolutionHamilton–Jacobi equationsintegro-differential equationscancerdrug resistance
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 377 - 399
Copyright
© EDP Sciences, SMAI, 2013

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

Références

Bacaër, N. and Sokhna, C., A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Math. Biosci. Eng. 2 (2005) 227–238. Google Scholar
G. Barles, Solutions de viscosité et équations de Hamilton–Jacobi. Collec. SMAI, Springer-Verlag, Paris (2002).
Barles, G. and Perthame, B., Concentrations and constrained Hamilton–Jacobi equations arising in adaptive dynamics, in Recent Developments in Nonlinear Partial Differential Equations, edited by D. Danielli. Contemp. Math. 439 (2007) 57–68. Google Scholar
Barles, G., Mirrahimi, S. and Perthame, B., Concentration in Lotka–Volterra parabolic or integral equations : a general convergence result. Methods Appl. Anal. 16 (2009) 321–340. Google Scholar
Bell, G. and Collins, S., Adaptation, extinction and global change. Evolutionary Applications 1 (2008) 3-16. Google ScholarPubMed
Bozic, I., Antal, T., Ohtsuki, H., Carter, H., Kim, D., Chen, S., Karchin, R., Kinzler, K.W., Vogelstein, B. and Nowak, M.A., Accumulation of driver and passenger mutations during tumor progression. Proc. Natl. Acad. Sci. USA 107 (2010) 18545–18550. Google ScholarPubMed
Calsina, À. and Cuadrado, S., A model for the adaptive dynamics of the maturation age. Ecol. Model. 133 (2000) 33–43. Google Scholar
Calsina, À. and Cuadrado, S., Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics. J. Math. Biol. 48 (2004) 135–159. Google Scholar
Champagnat, N., Ferrière, R. and Méléard, S., Unifying evolutionary dynamics : from individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69 (2006) 297–321. Google ScholarPubMed
Clairambault, J., Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments. Math. Model. Nat. Phenom. 4 (2009) 12–67 Google Scholar
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. Google Scholar
D’Agata, E.M.C., Dupont-Rouzeyrol, M., Magal, P., Olivier, D. and Ruan, S., The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria. PLoS One 3 (2008) es4036. Google ScholarPubMed
Day, T. and Bonduriansky, R., A unified approach to the evolutionary consequences of genetic and nongenetic inheritance. Amer. Nat. 178 (2011) E18–E36. Google ScholarPubMed
O. Diekmann, A beginner’s guide to adaptive dynamics, in Mathematical modeling of population dynamics, edited by R. Rudnicki. Banach Center Publications 63 (2004) 47–86.
Diekmann, O., Jabin, P.-E., Mischler, S. and Perthame, B., The dynamics of adaptation : an illuminating example and a Hamilton–Jacobi approach. Theor. Popul. Biol. 67 (2005) 257–271. Google Scholar
Fearon, E.R. and Vogelstein, B., A genetic model for colorectal tumorigenesis. Cell 61 (1990) 759–767. Google ScholarPubMed
Fleming, W.H. and Soner, H.M., Controlled markov processes and vicosity solutions. Appl. Math. 25 (1993). Google Scholar
Foo, J. and Michor, F., Evolution of resistance to targeted anti-cancer therapy during continuous and pulsed administration strategies. PLoS Comput. Biol. 5 (2009) e1000557. Google ScholarPubMed
Foo, J. and Michor, F., Evolution of resistance to anti-cancer therapy during general dosing schedules. J. Theor. Biol. 263 (2010) 179–188. Google ScholarPubMed
E.C. Friedberg, G.C. Walker, W. Siede, R.D. Wood, R.A. Schultz and T. Ellenberger, DNA repair and mutagenesis. ASM Press (2005).
Gatenby, R.A., A change of strategy in the war on cancer. Nature 459 (2009) 508–509. Google ScholarPubMed
Gatenby, R.A., Silva, A.S., Gillies, R.J. and Frieden, B.R., Adaptive therapy. Cancer Res. 69 (2009) 4894–4903. Google ScholarPubMed
J. Goldie and A. Coldman, Drug resistance in cancer : mechanisms and models. Cambridge University Press (1998).
Gomulkiewicz, R. and Holt, R.D., When does evolution by natural selection prevent extinction? Evolution 49 (1995) 201–207. Google ScholarPubMed
Gottesman, M.M., Fojo, T. and Bates, S.E., Multidrug resistance in cancer : role of ATP-dependent transporters. Nat. Rev. Cancer 2 (2002) 48–58. Google ScholarPubMed
Greaves, M. and Maley, C.C., Clonal evolution in cancer. Nature 481 (2012) 306–313. Google Scholar
Jabin, P.-E. and Raoul, G., Selection dynamics with competition. J. Math. Biol. 63 (2011) 493–517. Google Scholar
C.A. Jerez, Metal Extraction and Biomining, The Desk Encyclopedia of Microbiology, edited by M. Schaechter. Elsevier, Oxford 762–775.
Kimmel, M. and Świerniak, A., Control theory approach to cancer chemotherapy : benefiting from phase dependence and overcoming drug resistance, in Tutorials in Mathematical Biosciences III, edited by A. Friedman. Lect. Notes Math. 1872 (2006) 185–221. Google Scholar
Kivisaar, M., Stationary phase mutagenesis : mechanisms that accelerate adaptation of microbial populations under environmental stress. Environ. Microbiol. 5 (2003) 814–827. Google ScholarPubMed
Komarova, N.L. and Wodarz, D., Drug resistance in cancer : principles of emergence and prevention. Proc. Natl. Amer. Soc. 102 (2005) 9714–9719. Google Scholar
Lemesle, V., Mailleret, L. and Vaissayre, M., Role of spatial and temporal refuges in the evolution of pest resistance to toxic crops. Acta Biotheor. 58 (2010) 89–102. Google ScholarPubMed
Lorz, A., Mirrahimi, S. and Perthame, B., Dirac mass dynamics in multidimensional nonlocal parabolic equations. CPDE 36 (2011) 1071–1098. Google Scholar
Magal, P. and Mutation, Webb G.F., selection and recombination in a model of phenotype evolution. Discrete Contin. Dyn. Syst. 6 (2000) 221–236. Google Scholar
Marzac, C. et al., ATP-Binding-Cassette transporters associated with chemoresistance : transcriptional profiling in extreme cohorts and their prognostic impact in a cohort of 281 acute myeloid leukemia patients. Haematologica 96 (2011) 1293–1301. Google Scholar
McCormick, F., Cancer therapy based on oncogene addiction. J. Surg. Oncol. 103 (2011) 464–467. Google ScholarPubMed
Pasquier, J., Magal, P., Boulangé-Lecomte, C., Webb, G.F. and Le Foll, F., Consequences of cell-to-cell P-glycoprotein transfer on acquired multi-drug resistance in breast cancer : a cell population dynamics model. Biol. Direct 6 (2011) 5. Google ScholarPubMed
B. Perthame, Transport equations in biology. Series in Frontiers in Mathematics. Birkhauser (2007).
Perthame, B. and Barles, G., Dirac concentrations in Lotka–Volterra parabolic PDEs. Indiana Univ. Math. J. 57 (2008) 3275–3301. Google Scholar
Pienta, K.J., McGregor, N., Axelrod, R. and Axelrod, D.E., Ecological therapy for cancer : defining tumors using an ecosystem paradigm suggests new opportunities for novel cancer treatments. Transl. Oncol. 1 (2008) 158–164. Google ScholarPubMed
Rafii, A., Mirshahi, P., Poupot, M., Faussat, A.M., Simon, A., Ducros, E., Mery, E., Couderc, B., Lis, R., Capdet, J., Bergalet, J., Querleu, D., Dagonnet, F., Fournié, J.J., Marie, J.P., Pujade-Lauraine, E., Favre, G., Soria, J. and Mirshahi, M., Oncologic trogocytosis of an original stromal cells induces chemoresistance of ovarian tumours. PLoS One 3 (2008) e3894. Google ScholarPubMed
Scotto, K.W., Transcriptional regulation of ABC drug transporters. Oncogene 22 (2003) 7496–7511. Google ScholarPubMed
Shah, N.P., Tran, C.T., Lee, F.Y., Chan, P., Norris, D. and Sawyers, C.L., Overriding imatinib resistance with a novel ABL kinase inhibitor. Sci. Rep. 305 (2004) 399–401. Google ScholarPubMed
Silva, A.S. and Gatenby, R.A., A theoretical quantitative model for evolution of cancer chemotherapy resistance. Biol. Direct 5 (2010) 25. Google ScholarPubMed
Sprouffske, K., Pepper, J.W. and Maley, C.C., Accurate reconstruction of the temporal order of mutations in neoplastic progression. Cancer Prevention Res. 4 (2011) 1135–1144. Google ScholarPubMed
Terry, A.G. and Gourley, S.A., Perverse consequences of infrequently culling a pest. Bull. Math. Biol. 72 (2010) 1666–1695. Google Scholar
Tomasetti, C. and Levy, D. An elementary approach to modeling drug resistance in cancer. Math. Biosci. Eng. 7 (2010) 905–918. Google Scholar
C. Tomasetti and D. Levy, Drug resistance always depends on the cancer turnover rate, SBEC, in IFMBE Proc., edited by K.E. Herold, J. Vossoughi and W.E. Bentley. Springer, Berlin 32 (2010) 552–555.
Zhou, D.C., Ramond, S., Viguié, F., Faussat, A.-M., Zittoun, R. and Marie, J.-P., Sequential emergence of mrp and mdr-1 gene overexpression as well as mdr1-gene translocation in homoharringtonine selected K562 human leukemia cell lines. Int. J. Cancer 65 (1996) 365–371. Google Scholar