## PhD research topic - Corrado Mascia

#### Research areas

Biomathematics, differential equations, dynamical systems

#### Keywords

Mathematical modeling, Evolutive differential equations, Systems biology, Propagation fronts

#### Research topics

The project is devoted to the exploration of invasion fronts in tumor growth as described by Cancer Systems Biology language. The aim is to describe the phenomena of invasion of a state into another as a result of a behavior of competitive type, from both an analytical and numerical perspective. Stability analysis is also seen as a relevant issue.

The basic prototype is given by the standard scalar reaction-diffusion equation known as Fisher-KPP equation, composed by a standard diffusion second order operator coupled with a logistic-type reaction term. Such equation supports traveling waves connecting asymptotic states, describing the healthy and the sick states, respectively (using the language of epidemiology). More realistic models are given by systems of reaction-diffusion equations, incorporating, in some cases, possible degenerated terms. Other significant examples are epidemic fronts for competitive populations, and the Gatenby-Gawlinsky model describing the so-called Warburg effect. In addition, phenomena such as pulse propagation in (nerve/cardiac) fibers could be part of the analysis as in the realistic model introduced by Hodgkin-Huxley (with its simpler counterpart given by FitzHugh-Nagumo reaction-diffusion system). The topic is explored with the approach proper to Systems Biology paradigm, a biology-based interdisciplinary field of study that focuses on complex interactions within biological systems, comprising also the mathematical modeling.

The basic prototype is given by the standard scalar reaction-diffusion equation known as Fisher-KPP equation, composed by a standard diffusion second order operator coupled with a logistic-type reaction term. Such equation supports traveling waves connecting asymptotic states, describing the healthy and the sick states, respectively (using the language of epidemiology). More realistic models are given by systems of reaction-diffusion equations, incorporating, in some cases, possible degenerated terms. Other significant examples are epidemic fronts for competitive populations, and the Gatenby-Gawlinsky model describing the so-called Warburg effect. In addition, phenomena such as pulse propagation in (nerve/cardiac) fibers could be part of the analysis as in the realistic model introduced by Hodgkin-Huxley (with its simpler counterpart given by FitzHugh-Nagumo reaction-diffusion system). The topic is explored with the approach proper to Systems Biology paradigm, a biology-based interdisciplinary field of study that focuses on complex interactions within biological systems, comprising also the mathematical modeling.

Collaborations with the Working Group on Phase Transitions in Biology Through Mathematical Modeling, settled at the Systems Biology Group Lab – http://www.sbglab.org – Sapienza, University of Rome, are expected. As a last goal, such interaction should provide possible comparison with real experimental data.

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Bibliography

[1] McGillen J., Gaffney E.A., Martin N.K., Maini P.K., A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol. (2014) 68, 1199-1224.

[2] Noble D., Biophysics and systems biology, Phil. Trans. R. Soc. A (2010) 368, 1125-1139.

[3] Van Sarloos W., Front propagation into unstable states, Physics Reports 386 (2003) 29-222.