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**Growth of necrotic tumors in the presence and absence of inhibitors.**
*(English)*
Zbl 0856.92010

Summary: A mathematical model is presented for the growth of a multicellular spheroid that comprises a central core of necrotic cells surrounded by an outer annulus of proliferating cells. The model distinguishes two mechanisms for cell loss: apoptosis and necrosis. Cells loss due to apoptosis is defined to be programmed cell death, occurring, for example, when a cell exceeds its natural lifespan, whereas cell death due to necrosis is induced by changes in the cell’s microenvironment, occurring, for example, in nutrient-depleted regions.

Mathematically, the problem involves tracking two free boundaries, one for the outer tumor radius, the other for the inner necrotic radius. Numerical simulations of the model are presented in an inhibitor-free setting and an inhibitor-present setting for various parameter values. The effects of nutrients and inhibitors on the existence and stability of the time-independent solutions of the model are studied using a combination of numerical and asymptotic techniques.

Mathematically, the problem involves tracking two free boundaries, one for the outer tumor radius, the other for the inner necrotic radius. Numerical simulations of the model are presented in an inhibitor-free setting and an inhibitor-present setting for various parameter values. The effects of nutrients and inhibitors on the existence and stability of the time-independent solutions of the model are studied using a combination of numerical and asymptotic techniques.

### MSC:

92C50 | Medical applications (general) |

35R35 | Free boundary problems for PDEs |

35K57 | Reaction-diffusion equations |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

65Z05 | Applications to the sciences |

### Keywords:

cancer growth; growth of multicellular spheroids; necrotic cells; proliferating cells; cell loss; apoptosis; necrosis; inhibitor-free setting; inhibitor-present setting; nutrients; existence; stability; time-independent solutions
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\textit{H. M. Byrne} and \textit{M. A. J. Chaplain}, Math. Biosci. 135, No. 2, 187--216 (1996; Zbl 0856.92010)

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### References:

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