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

*(English)*Zbl 0836.92011Summary: A model for the evolution of a spherically symmetric, nonnecrotic tumor is presented. The effects of nutrients and inhibitors on the existence and stability of time-independent solutions are studied. With a single nutrient and no inhibitors present, the trivial solution, which corresponds to a state in which no tumor is present, persists for all parameter values, whereas the nontrivial solution, which corresponds to a tumor of finite size, exists for only a prescribed range of parameters, which corresponds to a balance between cell proliferation and cell death.

Stability analysis, based on a two-timing method, suggests that, where it exists, the nontrivial solution is stable and the trivial solution unstable. Otherwise, the trivial solution is stable. Modifications to these characteristic states brought about by the presence of different types of inhibitors are also investigated and shown to have significant effect. Implications of the model for the treatment of cancer are also discussed.

Stability analysis, based on a two-timing method, suggests that, where it exists, the nontrivial solution is stable and the trivial solution unstable. Otherwise, the trivial solution is stable. Modifications to these characteristic states brought about by the presence of different types of inhibitors are also investigated and shown to have significant effect. Implications of the model for the treatment of cancer are also discussed.

##### MSC:

92C50 | Medical applications (general) |

##### Keywords:

spherically symmetric nonnecrotic tumor; nutrients; inhibitors; existence; stability; time-independent solutions; cell proliferation; cell death; two-timing method; treatment of cancer
PDF
BibTeX
XML
Cite

\textit{H. M. Byrne} and \textit{M. A. J. Chaplain}, Math. Biosci. 130, No. 2, 151--181 (1995; Zbl 0836.92011)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Adam, J.A., A simplified mathematical model of tumor growth, Math. biosci., 81, 229-244, (1986) · Zbl 0601.92007 |

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

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

[4] | Adam, J.A., Corrigendum: A mathematical model of tumor growth by diffusion, Math. biosci., 94, 155, (1989) |

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

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

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

[8] | 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) |

[9] | 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) |

[10] | 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) |

[11] | Sutherland, R.M.; Durand, R.E., Radiation response of multicell spheroids: an in vitro tumour model, Internat. J. radiat. biol., 23, 235-246, (1973) |

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

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

[14] | Chaplain, M.A.J., Mathematical models for the growth of solid tumours and the tip morphogenesis of acetabularia, () · Zbl 0859.92012 |

[15] | Folkman, J., Tumor angiogenesis, Adv. cancer res., 19, 331-338, (1974) |

[16] | Folkman, J., The vascularization of tumors, (), 115-124 |

[17] | Kerr, J.F.R., Shrinkage necrosis: A distinct mode of cellular death, J. path., 105, 13-20, (1971) |

[18] | 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, 25, 239-257, (1972) |

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

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

[21] | Jordan, D.W.; Smith, P., Nonlinear ordinary differential equations, (1977), Clarendon Press Oxford, UK · Zbl 0417.34002 |

[22] | Nayfeh, A.H., Perturbation methods, (1973), John Wiley and Sons, Inc · Zbl 0375.35005 |

[23] | Chaplain, M.A.J.; Britton, N.F., On the concentration profile of a growth inhibitory factor in multicell spheroids, Math. biosci., 115, 233-245, (1993) · Zbl 0771.92009 |

[24] | Smith, G.D., Numerical solution of partial differential equations: finite difference methods, (1985), Clarendon Press · Zbl 0576.65089 |

[25] | McElwain, D.L.S.; Pettet, G., Cell migration in multicell spheroids: swimming against the tide, Bull. math. biol., 55, 655-674, (1993) · Zbl 0765.92015 |

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

[27] | Freyer, J.P.; Tustanoff, E.; Franko, A.J.; Sutherland, R.M., In situ oxygen consumption rates of cells in V-79 multicellular spheroids during growth, J. cell. physiol., 118, 53-61, (1984) |

[28] | Freyer, J.P.; Schor, P.L., Regrowth of cells from multicell tumour spheroids, Cell tissue kinet., 20, 249, (1987) |

[29] | Grobe, K.; Mueller-Klieser, W., Distributions of oxygen, nutrient, and metabolic waste concentrations in multicellular spheroids and their dependence on spheroid parameters, Eur. biophys. J., 19, 169-181, (1991) |

[30] | Landry, J.; Freyer, J.P.; Sutherland, A.M., A model for the growth of multicell spheroids, Cell tissue kinet., 15, 585-594, (1982) |

[31] | Mueller-Klieser, W., Multicellular spheroids: A review on cellular aggregates in cancer research, J. cancer res. clin. oncol., 113, 101-122, (1987) |

[32] | Sutherland, R.M.; McCredie, J.A.; Inch, W.R., Growth of multicell spheroids as a model of nodular carcinomas, J. nat. cancer inst., 46, 113-120, (1971) |

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

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

[35] | Wibe, E.; Lindmo, T.; Kaalhus, O., Cell kinetic characteristics in different parts of multicellular spheroids of human origin, Cell tissue kinet., 14, 639-651, (1981) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.