Tumor growth dynamics with nutrient limitation and cell proliferation time delay.

*(English)*Zbl 1375.34119Summary: It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

##### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K20 | Stability theory of functional-differential equations |

92C50 | Medical applications (general) |

92C37 | Cell biology |

##### Keywords:

nonlinear tumor model; nutrient limitation; time delay; quiescence; proliferation; stability
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\textit{A. Alsheri} et al., Discrete Contin. Dyn. Syst., Ser. B 22, No. 10, 3771--3782 (2017; Zbl 1375.34119)

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##### References:

[1] | J. A. Adam, <em>A Survey of Models on Tumour Immune Systems Dynamics</em>,, Birkhüauser, (1997) |

[2] | E. O. Alzahrani, Quiescence as an explanation of Gompertzian tumor growth revisited,, Mathematical Biosciences, 254, 76, (2014) · Zbl 1325.92041 |

[3] | E. O. Alzahrani, Nutrient limitations as an explanation of Gompertzian tumor growth,, Discrete Cont. Dyn. Syst.-B., 21, 357, (2016) · Zbl 1343.92206 |

[4] | R. P. Araujo, A history of the study of solid tumour growth: The contribution of mathematical modelling,, Bull. Math. Biol., 66, 1039, (2004) · Zbl 1334.92187 |

[5] | L. von Bertalanffy, Quantitative laws in metabolism and growth,, Quart. Rev. Biol., 32, 217, (1957) |

[6] | H. M. Byrne, A two-phase model of solid tumour growth,, Applied Mathematics Letters, 16, 567, (2003) · Zbl 1040.92015 |

[7] | S. E. Eikenberry, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model,, Biology Direct, 5, (2010) |

[8] | S. E. Eikenberry, Tumor-immune interaction, surgical treatment and cancer recurrence in a mathematical model of melanoma,, PLoS Comput. Biol., 5, (2009) |

[9] | R. A. Everett, Data and implication based comparison of two chronic myeloid leukemia models,, Math. Biosc. Eng., 10, 1501, (2013) · Zbl 1273.92026 |

[10] | C. L. Frenzen, A cell kinetics justification for Gompertz equation,, SIAM J. Appl. Math., 46, 614, (1986) · Zbl 0608.92012 |

[11] | S. Geritz, A mechanistic derivation of the DeAngelis-Beddington functional response,, Journal of Theoretical Biology, 314, 106, (2012) |

[12] | B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies,, Phil. Trans. Royal Soc. London, 115, 513, (1825) |

[13] | M. Gyllenberg, Quiescence as an explanation of Gompertzian tumor growth,, Growth Dev Aging, 53, 25, (1989) |

[14] | M. Gyllenberg, A nonlinear structured population model of tumor growth with quiescence,, J. Math. Biol., 28, 671, (1990) · Zbl 0744.92026 |

[15] | F. Kozusko, Combining Gompertzian growth and cell population dynamics,, Math. Biosc., 185, 153, (2003) · Zbl 1021.92012 |

[16] | Y. Kuang, <em>Delay Differential Equations: With Applications in Population Dynamics</em>,, Academic Press, (1993) · Zbl 0777.34002 |

[17] | Y. Kuang, <em>Introduction to Mathematical Oncology</em>,, CRC Press, (2016) · Zbl 1341.92002 |

[18] | A. O. Martinez, Growth dynamics of multicell spheroids from three murine tumors,, Growth, 44, 112, (1980) |

[19] | M. Marusic, Prediction power of mathematical models for tumor growth,, Journal of Biological Systems, 1, 69, (1993) |

[20] | L. Norton, Predicting the course of Gompertzian growth,, Nature, 264, 542, (1976) |

[21] | T. Portz, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy,, AIP Advances, 2, (2012) |

[22] | J. A. Sherratt, A new mathematical model for avascular tumor growth,, J. Math. Biol., 43, 291, (2001) · Zbl 0990.92021 |

[23] | C. J. Thalhauser, Explicit separation of growth and motility in a new tumor cord model,, Bulletin of Math. Biol., 71, 585, (2009) · Zbl 1163.92018 |

[24] | D. Wallace, Properties of tumor spheroid growth exhibited by simple mathematical models,, Frontiers in Oncology, 3, 1, (2013) |

[25] | R. Yafia, Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescence,, Nonlinear Analysis: Modelling and Control, 11, 95, (2006) · Zbl 1107.92025 |

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.