Nutrient limitations as an explanation of Gompertzian tumor growth

Ebraheem O.  Alzahrani; Yang Kuang

doi:10.3934/dcdsb.2016.21.357

Article Contents

2016, Volume 21, Issue 2: 357-372. Doi: 10.3934/dcdsb.2016.21.357

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Nutrient limitations as an explanation of Gompertzian tumor growth

Ebraheem O. Alzahrani^1, and
Yang Kuang^2,

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589
2.
School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281

Received: December 31, 2014

Revised: March 31, 2015

Published: February 2016

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Abstract

An intuitive and influential two-compartment model of cancer cell growth proposed by Gyllenberg and Webb in 1989 [18] with transition rates between proliferating and quiescent cells reproduces some important features of the well known Gompertzian growth model. However, other plausible mechanisms may also be capable of producing similar dynamics. In this paper, we formulate a resource limited three-compartment model of avascular spherical solid tumor growth and study its dynamics. The resource, such as oxygen, is assumed to enter the tumor proportional to its surface area and the dead cells form the necrotic core inside the tumor. We show the tumor growth of our model mimics that of Gompertzian model, and the solutions of our model are naturally bounded. We also identify general and explicit expressions of the tumor final sizes and study the stability of the tumor at steady states. In contrast to the Gyllenberg-Webb model, our model confirms that tumor size at the positive steady state is strictly decreasing function of the dead cell removal rate. We also present two intriguing mathematical open questions.

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Mathematics Subject Classification: Primary: 34K20, 92C50; Secondary: 92D25.

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