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Some mathematical aspects of tumor growth and therapy. (English) Zbl 1373.35322
Jang, Sun Young (ed.) et al., Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. I: Plenary lectures and ceremonies. Seoul: KM Kyung Moon Sa (ISBN 978-89-6105-804-9/hbk; 978-89-6105-803-2/set). 529-545 (2014).
Summary: Mathematical models of tumor growth, written as partial differential equations or free boundary problems, are now in the toolbox for predicting the evolution of some cancers, using model based image analysis for example. These models serve not only to predict the evolution of cancers in medical treatments but also to understand the biological and mechanical effects that are involved in the tissue growth, the optimal therapy and, in some cases, in their implication in therapeutic failures. The models under consideration contain several levels of complexity, both in terms of the biological and mechanical effects, and therefore in their mathematical description. The number of scales, from the molecules, to the cell, to the organ and the entire body, explains partly the complexity of the problem. This paper focusses on two aspects of the problem which can be described with mathematical models keeping some simplicity. They have been chosen so as to cover mathematical questions which stem from both mechanical laws and biological considerations. I shall first present an asymptotic problem describing some mechanical properties of tumor growth and secondly, models of resistance to therapy and cell adaptation again using asymptotic analysis.
For the entire collection see [Zbl 1314.00103].

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K55 Nonlinear parabolic equations
35B25 Singular perturbations in context of PDEs
76D27 Other free boundary flows; Hele-Shaw flows
92C50 Medical applications (general)
92D25 Population dynamics (general)
35Q35 PDEs in connection with fluid mechanics