Growth of necrotic tumors in the presence and absence of inhibitors.

*(English)*Zbl 0856.92010Summary: 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:

[1] | Greenspan, H.P., Models for the growth of a solid tumour by diffusion, Stud. appl. math., 52, 317-340, (1972) · Zbl 0257.92001 |

[2] | McElwain, D.L.S.; Ponzo, P.J., A model for the growth of a solid tumor with non-uniform oxygen consumption, Math. biosci., 35, 267-279, (1977) · Zbl 0364.92020 |

[3] | Adam, J.A., A mathematical model of tumour growth. II. effects of geometry and spatial uniformity on stability, Math. biosci., 86, 183-211, (1987) · Zbl 0634.92002 |

[4] | Adam, J.A., A mathematical model of tumor growth. III. comparison with experiment, Math. biosci., 86, 213-227, (1987) · Zbl 0634.92003 |

[5] | Adam, J.A.; Maggelakis, S.A., Diffusion regulated growth characteristics of a spherical prevascular carcinoma, Bull. math. biol., 52, 549-582, (1990) · Zbl 0712.92010 |

[6] | Maggelakis, S.A.; Adam, J.A., Mathematical model of a prevascular growth of a spherical carcinoma, Math. comput. modelling, 13, 23-38, (1990) · Zbl 0706.92010 |

[7] | Chaplain, M.A.J.; Britton, N.F., On the concentration profile of a growth inhibitory factor in multicell spheroids, Math. biosci., 115, 223-245, (1993) · Zbl 0771.92009 |

[8] | Marusic, M.; Bajzer, Z.; Freyer, J.P.; Vuk-Pavlovic, S., Analysis of growth of multicellular tumour spheroids by mathematical models, Cell prolif., 27, 73-94, (1994) |

[9] | Tubiana, M., The kinetics of tumour cell proliferation and radiotherapy, Br. J. radiol., 44, 325-347, (1971) |

[10] | Sutherland, R.M.; Durand, R.E., Radiation response of multicell spheroidsâ€”an in vitro tumour model, Int. J. radiat. biol., 23, 235-246, (1973) |

[11] | Sutherland, R.M.; Durand, R.E., Growth and cellular characteristics of multicell spheroids, Recent results cancer res., 95, 24-49, (1984) |

[12] | Freyer, J.P.; Sutherland, R.M., Regulation of growth saturation and development of necrosis in EMT6/ro multicellular spheroids by the glucose and oxygen supply, Cancer res., 46, 3504-3512, (1986) |

[13] | Freyer, J.P.; Sutherland, R.M., Proliferative and clonogenic heterogeneity of cells from EMT6/ro multicellular spheroids induced by the glucose and oxygen supply, Cancer res., 46, 3513-3520, (1986) |

[14] | Sutherland, R.M., Cell and environment interactions in tumor microregions: the multicell spheroid model, Science, 240, 177-184, (1988) |

[15] | Adam, G.; Steiner, U.; Maier, H.; Ulrich, S., Analysis of cellular interactions in density-dependent inhibition of 3T3 cell proliferation, Biophys. struct. mech., 9, 75-82, (1982) |

[16] | Iversen, O.H., Epidermal chalones and squamous cell carcinomas, Virchows arch. B cell pathol., 27, 229-235, (1978) |

[17] | Iversen, O.H., The chalones, (), 491-550 |

[18] | Iversen, O.H., What’s new in endogenous growth stimulators and inhibitors (chalones), Pathol. res. pract., 180, 77-80, (1985) |

[19] | Byrne, H.M.; Chaplain, M.A.J., Growth of non-necrotic tumours in the presence and absence of inhibitors, Math. biosci., 130, 151-181, (1995) · Zbl 0836.92011 |

[20] | Kerr, J.F.R., Shrinkage necrosis; a distinct mode of cellular death, J. pathol., 105, 13-20, (1971) |

[21] | Kerr, J.F.R.; Wyllie, A.H.; Currie, A.R., Apoptosis: a basic biological phenomenon with wide ranging implications in tissue kinetics, Br. J. cancer, 26, 239-257, (1972) |

[22] | Darzynkiewicz; Bruno, S.; del Bino, G.; Gorczyca, W.; Hotz, M.A.; Lassota, P.; Traganos, F., Features of apoptotic cells measured by flow cytometry, Cytometry, 13, 795-808, (1992) |

[23] | McElwain, D.L.S.; Morris, L.E., Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth, Math. biosci., 39, 147-157, (1978) |

[24] | Durand, R.E., Cell cycle kinetics in an in vitro tumour model, Cell tissue kinet., 9, 403-412, (1976) |

[25] | Durand, R.E., Multicell spheroids as a model for cell kinetic studies, Cell tissue kinet., 23, 141-159, (1990) |

[26] | Folkman, J.; Hochberg, M., Self-regulation of growth in three dimensions, J. exp. med., 138, 745-753, (1973) |

[27] | Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1984), Springer-Verlag New York · Zbl 0153.13602 |

[28] | Byrne, H.M.; Chaplain, M.A.J., The role of cell-cell adhesion in the growth & development of carcinomas, (1996), (submitted) · Zbl 0883.92014 |

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