# Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies

Alexander Lorz; Tommaso Lorenzi; Michael E. Hochberg; Jean Clairambault; Benoît Perthame

- Volume: 47, Issue: 2, page 377-399
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topLorz, Alexander, et al. "Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 377-399. <http://eudml.org/doc/273323>.

@article{Lorz2013,

abstract = {Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous 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 of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton–Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.},

author = {Lorz, Alexander, Lorenzi, Tommaso, Hochberg, Michael E., Clairambault, Jean, Perthame, Benoît},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {mathematical oncology; adaptive evolution; Hamilton–Jacobi equations; integro-differential equations; cancer; drug resistance; Hamilton-Jacobi equations},

language = {eng},

number = {2},

pages = {377-399},

publisher = {EDP-Sciences},

title = {Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies},

url = {http://eudml.org/doc/273323},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Lorz, Alexander

AU - Lorenzi, Tommaso

AU - Hochberg, Michael E.

AU - Clairambault, Jean

AU - Perthame, Benoît

TI - Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 2

SP - 377

EP - 399

AB - Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous 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 of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton–Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.

LA - eng

KW - mathematical oncology; adaptive evolution; Hamilton–Jacobi equations; integro-differential equations; cancer; drug resistance; Hamilton-Jacobi equations

UR - http://eudml.org/doc/273323

ER -

## References

top- [1] N. Bacaër and C. Sokhna, A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Math. Biosci. Eng.2 (2005) 227–238. Zbl1070.92034MR2144237
- [2] G. Barles, Solutions de viscosité et équations de Hamilton–Jacobi. Collec. SMAI, Springer-Verlag, Paris (2002). Zbl0819.35002
- [3] G. Barles and B. Perthame, 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. Zbl1137.49027MR2359020
- [4] G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka–Volterra parabolic or integral equations : a general convergence result. Methods Appl. Anal.16 (2009) 321–340. Zbl1204.35027MR2650800
- [5] G. Bell and S. Collins, Adaptation, extinction and global change. Evolutionary Applications1 (2008) 3-16.
- [6] I. Bozic, T. Antal, H. Ohtsuki, H. Carter, D. Kim, S. Chen, R. Karchin, K.W. Kinzler, B. Vogelstein and M.A. Nowak, Accumulation of driver and passenger mutations during tumor progression. Proc. Natl. Acad. Sci. USA107 (2010) 18545–18550.
- [7] À. Calsina and S. Cuadrado, A model for the adaptive dynamics of the maturation age. Ecol. Model.133 (2000) 33–43.
- [8] À. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics. J. Math. Biol.48 (2004) 135–159. Zbl1078.92051MR2041498
- [9] N. Champagnat, R. Ferrière and S. Méléard, Unifying evolutionary dynamics : from individual stochastic processes to macroscopic models. Theor. Popul. Biol.69 (2006) 297–321. Zbl1118.92039
- [10] J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments. Math. Model. Nat. Phenom.4 (2009) 12–67 Zbl1165.92021MR2519055
- [11] M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc.27 (1992) 1–67. Zbl0755.35015MR1118699
- [12] E.M.C. D’Agata, M. Dupont-Rouzeyrol, P. Magal, D. Olivier and S. Ruan, The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria. PLoS One 3 (2008) es4036.
- [13] T. Day and R. Bonduriansky, A unified approach to the evolutionary consequences of genetic and nongenetic inheritance. Amer. Nat. 178 (2011) E18–E36.
- [14] 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. Zbl1051.92032MR2076953
- [15] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation : an illuminating example and a Hamilton–Jacobi approach. Theor. Popul. Biol.67 (2005) 257–271. Zbl1072.92035
- [16] E.R. Fearon and B. Vogelstein, A genetic model for colorectal tumorigenesis. Cell61 (1990) 759–767.
- [17] W.H. Fleming and H.M. Soner, Controlled markov processes and vicosity solutions. Appl. Math. 25 (1993). Zbl0773.60070MR1199811
- [18] J. Foo and F. Michor, Evolution of resistance to targeted anti-cancer therapy during continuous and pulsed administration strategies. PLoS Comput. Biol. 5 (2009) e1000557. MR2577428
- [19] J. Foo and F. Michor, Evolution of resistance to anti-cancer therapy during general dosing schedules. J. Theor. Biol.263 (2010) 179–188. MR2980705
- [20] E.C. Friedberg, G.C. Walker, W. Siede, R.D. Wood, R.A. Schultz and T. Ellenberger, DNA repair and mutagenesis. ASM Press (2005).
- [21] R.A. Gatenby, A change of strategy in the war on cancer. Nature459 (2009) 508–509.
- [22] R.A. Gatenby, A.S. Silva, R.J. Gillies and B.R. Frieden, Adaptive therapy. Cancer Res.69 (2009) 4894–4903.
- [23] J. Goldie and A. Coldman, Drug resistance in cancer : mechanisms and models. Cambridge University Press (1998). Zbl0519.92008
- [24] R. Gomulkiewicz and R.D. Holt, When does evolution by natural selection prevent extinction? Evolution49 (1995) 201–207.
- [25] M.M. Gottesman, T. Fojo and S.E. Bates, Multidrug resistance in cancer : role of ATP-dependent transporters. Nat. Rev. Cancer2 (2002) 48–58.
- [26] M. Greaves and C.C. Maley, Clonal evolution in cancer. Nature481 (2012) 306–313.
- [27] P.-E. Jabin and G. Raoul, Selection dynamics with competition. J. Math. Biol.63 (2011) 493–517. Zbl1230.92038MR2824978
- [28] C.A. Jerez, Metal Extraction and Biomining, The Desk Encyclopedia of Microbiology, edited by M. Schaechter. Elsevier, Oxford 762–775.
- [29] M. Kimmel and A. Świerniak, 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. MR2216629
- [30] M. Kivisaar, Stationary phase mutagenesis : mechanisms that accelerate adaptation of microbial populations under environmental stress. Environ. Microbiol.5 (2003) 814–827.
- [31] N.L. Komarova and D. Wodarz, Drug resistance in cancer : principles of emergence and prevention. Proc. Natl. Amer. Soc.102 (2005) 9714–9719.
- [32] V. Lemesle, L. Mailleret and M. Vaissayre, Role of spatial and temporal refuges in the evolution of pest resistance to toxic crops. Acta Biotheor.58 (2010) 89–102.
- [33] A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations. CPDE36 (2011) 1071–1098. Zbl1229.35113MR2765430
- [34] P. Magal and Webb G.F.Mutation, selection and recombination in a model of phenotype evolution. Discrete Contin. Dyn. Syst.6 (2000) 221–236. Zbl1007.92027MR1739925
- [35] C. Marzac 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. Haematologica96 (2011) 1293–1301.
- [36] F. McCormick, Cancer therapy based on oncogene addiction. J. Surg. Oncol.103 (2011) 464–467.
- [37] J. Pasquier, P. Magal, C. Boulangé-Lecomte, G.F. Webb and F. Le Foll, 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.
- [38] B. Perthame, Transport equations in biology. Series in Frontiers in Mathematics. Birkhauser (2007). Zbl1185.92006MR2270822
- [39] B. Perthame and G. Barles, Dirac concentrations in Lotka–Volterra parabolic PDEs. Indiana Univ. Math. J.57 (2008) 3275–3301. Zbl1172.35005MR2492233
- [40] K.J. Pienta, N. McGregor, R. Axelrod and D.E. Axelrod, Ecological therapy for cancer : defining tumors using an ecosystem paradigm suggests new opportunities for novel cancer treatments. Transl. Oncol.1 (2008) 158–164.
- [41] A. Rafii, P. Mirshahi, M. Poupot, A.M. Faussat, A. Simon, E. Ducros, E. Mery, B. Couderc, R. Lis, J. Capdet, J. Bergalet, D. Querleu, F. Dagonnet, J.J. Fournié, J.P. Marie, E. Pujade-Lauraine, G. Favre, J. Soria and M. Mirshahi, Oncologic trogocytosis of an original stromal cells induces chemoresistance of ovarian tumours. PLoS One 3 (2008) e3894.
- [42] K.W. Scotto, Transcriptional regulation of ABC drug transporters. Oncogene22 (2003) 7496–7511.
- [43] N.P. Shah, C.T. Tran, F.Y. Lee, P. Chan, D. Norris and C.L. Sawyers, Overriding imatinib resistance with a novel ABL kinase inhibitor. Sci. Rep.305 (2004) 399–401.
- [44] A.S. Silva and R.A. Gatenby, A theoretical quantitative model for evolution of cancer chemotherapy resistance. Biol. Direct 5 (2010) 25.
- [45] K. Sprouffske, J.W. Pepper and C.C. Maley, Accurate reconstruction of the temporal order of mutations in neoplastic progression. Cancer Prevention Res.4 (2011) 1135–1144.
- [46] A.G. Terry and S.A. Gourley, Perverse consequences of infrequently culling a pest. Bull. Math. Biol.72 (2010) 1666–1695. Zbl1201.92066MR2728001
- [47] Tomasetti C. and Levy D.An elementary approach to modeling drug resistance in cancer. Math. Biosci. Eng.7 (2010) 905–918. Zbl1259.92056MR2739996
- [48] 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.
- [49] D.C. Zhou, S. Ramond, F. Viguié, A.-M. Faussat, R. Zittoun and J.-P. Marie, 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. Cancer65 (1996) 365–371.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.