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.
Keywords: mathematical oncology, adaptive evolution, Hamilton-Jacobi equations, integro-differential equations, cancer, drug resistance
@article{M2AN_2013__47_2_377_0, author = {Lorz, Alexander and Lorenzi, Tommaso and Hochberg, Michael E. and Clairambault, Jean and Perthame, Beno{\^\i}t}, title = {Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {377--399}, publisher = {EDP-Sciences}, volume = {47}, number = {2}, year = {2013}, doi = {10.1051/m2an/2012031}, mrnumber = {3021691}, zbl = {1274.92025}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012031/} }
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 PY - 2013 SP - 377 EP - 399 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012031/ DO - 10.1051/m2an/2012031 LA - en ID - M2AN_2013__47_2_377_0 ER -
%0 Journal Article %A Lorz, Alexander %A Lorenzi, Tommaso %A Hochberg, Michael E. %A Clairambault, Jean %A Perthame, Benoît %T Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 377-399 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012031/ %R 10.1051/m2an/2012031 %G en %F M2AN_2013__47_2_377_0
Lorz, Alexander; Lorenzi, Tommaso; Hochberg, Michael E.; Clairambault, Jean; Perthame, Benoît. Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 2, pp. 377-399. doi : 10.1051/m2an/2012031. http://www.numdam.org/articles/10.1051/m2an/2012031/
[1] A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Math. Biosci. Eng. 2 (2005) 227-238. | MR | Zbl
and ,[2] Solutions de viscosité et équations de Hamilton-Jacobi. Collec. SMAI, Springer-Verlag, Paris (2002). | Zbl
,[3] 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. | MR | Zbl
and ,[4] Concentration in Lotka-Volterra parabolic or integral equations : a general convergence result. Methods Appl. Anal. 16 (2009) 321-340. | MR | Zbl
, and ,[5] Adaptation, extinction and global change. Evolutionary Applications 1 (2008) 3-16.
and ,[6] Accumulation of driver and passenger mutations during tumor progression. Proc. Natl. Acad. Sci. USA 107 (2010) 18545-18550.
, , , , , , , , and ,[7] A model for the adaptive dynamics of the maturation age. Ecol. Model. 133 (2000) 33-43.
and ,[8] Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics. J. Math. Biol. 48 (2004) 135-159. | MR | Zbl
and ,[9] Unifying evolutionary dynamics : from individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69 (2006) 297-321. | Zbl
, and ,[10] Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments. Math. Model. Nat. Phenom. 4 (2009) 12-67 | MR | Zbl
,[11] User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR | Zbl
, and ,[12] The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria. PLoS One 3 (2008) es4036.
, , , and ,[13] A unified approach to the evolutionary consequences of genetic and nongenetic inheritance. Amer. Nat. 178 (2011) E18-E36.
and ,[14] A beginner's guide to adaptive dynamics, in Mathematical modeling of population dynamics, edited by R. Rudnicki. Banach Center Publications 63 (2004) 47-86. | MR | Zbl
,[15] The dynamics of adaptation : an illuminating example and a Hamilton-Jacobi approach. Theor. Popul. Biol. 67 (2005) 257-271. | Zbl
, , and ,[16] A genetic model for colorectal tumorigenesis. Cell 61 (1990) 759-767.
and ,[17] Controlled markov processes and vicosity solutions. Appl. Math. 25 (1993). | MR | Zbl
and ,[18] Evolution of resistance to targeted anti-cancer therapy during continuous and pulsed administration strategies. PLoS Comput. Biol. 5 (2009) e1000557. | MR
and ,[19] Evolution of resistance to anti-cancer therapy during general dosing schedules. J. Theor. Biol. 263 (2010) 179-188. | MR
and ,[20] DNA repair and mutagenesis. ASM Press (2005).
, , , , and ,[21] A change of strategy in the war on cancer. Nature 459 (2009) 508-509.
,[22] Adaptive therapy. Cancer Res. 69 (2009) 4894-4903.
, , and ,[23] Drug resistance in cancer : mechanisms and models. Cambridge University Press (1998).
and ,[24] When does evolution by natural selection prevent extinction? Evolution 49 (1995) 201-207.
and ,[25] Multidrug resistance in cancer : role of ATP-dependent transporters. Nat. Rev. Cancer 2 (2002) 48-58.
, and ,[26] Clonal evolution in cancer. Nature 481 (2012) 306-313.
and ,[27] Selection dynamics with competition. J. Math. Biol. 63 (2011) 493-517. | MR | Zbl
and ,[28] Metal Extraction and Biomining, The Desk Encyclopedia of Microbiology, edited by M. Schaechter. Elsevier, Oxford 762-775.
,[29] 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. | MR
and ,[30] Stationary phase mutagenesis : mechanisms that accelerate adaptation of microbial populations under environmental stress. Environ. Microbiol. 5 (2003) 814-827.
,[31] Drug resistance in cancer : principles of emergence and prevention. Proc. Natl. Amer. Soc. 102 (2005) 9714-9719.
and ,[32] Role of spatial and temporal refuges in the evolution of pest resistance to toxic crops. Acta Biotheor. 58 (2010) 89-102.
, and ,[33] Dirac mass dynamics in multidimensional nonlocal parabolic equations. CPDE 36 (2011) 1071-1098. | MR | Zbl
, and ,[34] Mutation, selection and recombination in a model of phenotype evolution. Discrete Contin. Dyn. Syst. 6 (2000) 221-236. | MR | Zbl
and[35] 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.
et al.,[36] Cancer therapy based on oncogene addiction. J. Surg. Oncol. 103 (2011) 464-467.
,[37] 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.
, , , and ,[38] Transport equations in biology. Series in Frontiers in Mathematics. Birkhauser (2007). | MR | Zbl
,[39] Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ. Math. J. 57 (2008) 3275-3301. | MR | Zbl
and ,[40] Ecological therapy for cancer : defining tumors using an ecosystem paradigm suggests new opportunities for novel cancer treatments. Transl. Oncol. 1 (2008) 158-164.
, , and ,[41] Oncologic trogocytosis of an original stromal cells induces chemoresistance of ovarian tumours. PLoS One 3 (2008) e3894.
, , , , , , , , , , , , , , , , , and ,[42] Transcriptional regulation of ABC drug transporters. Oncogene 22 (2003) 7496-7511.
,[43] Overriding imatinib resistance with a novel ABL kinase inhibitor. Sci. Rep. 305 (2004) 399-401.
, , , , and ,[44] A theoretical quantitative model for evolution of cancer chemotherapy resistance. Biol. Direct 5 (2010) 25.
and ,[45] Accurate reconstruction of the temporal order of mutations in neoplastic progression. Cancer Prevention Res. 4 (2011) 1135-1144.
, and ,[46] Perverse consequences of infrequently culling a pest. Bull. Math. Biol. 72 (2010) 1666-1695. | MR | Zbl
and ,[47] An elementary approach to modeling drug resistance in cancer. Math. Biosci. Eng. 7 (2010) 905-918. | MR | Zbl
and[48] 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.
and ,[49] 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.
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