Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, pp. 377-399.

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.

DOI: 10.1051/m2an/2012031
Classification: 35B25,  45M05,  49L25,  92C50,  92D15
Keywords: mathematical oncology, adaptive evolution, Hamilton-Jacobi equations, integro-differential equations, cancer, drug resistance
     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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {377--399},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {2},
     year = {2013},
     doi = {10.1051/m2an/2012031},
     zbl = {1274.92025},
     mrnumber = {3021691},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012031/}
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
DA  - 2013///
SP  - 377
EP  - 399
VL  - 47
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012031/
UR  - https://zbmath.org/?q=an%3A1274.92025
UR  - https://www.ams.org/mathscinet-getitem?mr=3021691
UR  - https://doi.org/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 - Modélisation Mathématique et Analyse Numérique
%D 2013
%P 377-399
%V 47
%N 2
%I EDP-Sciences
%U https://doi.org/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 - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, pp. 377-399. doi : 10.1051/m2an/2012031. http://www.numdam.org/articles/10.1051/m2an/2012031/

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