Tumor growth dynamics with nutrient limitation and cell proliferation time delay

Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky, Yang Kuang

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)3771-3782
Number of pages12
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume22
Issue number10
DOIs
StatePublished - Dec 1 2017

Fingerprint

Cell Proliferation
Tumor Growth
Cell proliferation
Nutrients
Tumors
Time Delay
Time delay
Tumor
Cell
Cell growth
Tend
Limiting
Compartment Model
Compartmental Model
Oscillatory Solution
Time and motion study
Characteristic equation
Zero
Population Model
Surface area

Keywords

  • Nonlinear tumor model
  • Nutrient limitation
  • Proliferation
  • Quiescence
  • Stability
  • Time delay

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Tumor growth dynamics with nutrient limitation and cell proliferation time delay. / Alsheri, Ahuod; Alzahrani, Ebraheem O.; Asiri, Asim; El-Dessoky, Mohamed M.; Kuang, Yang.

In: Discrete and Continuous Dynamical Systems - Series B, Vol. 22, No. 10, 01.12.2017, p. 3771-3782.

Research output: Contribution to journalArticle

Alsheri, Ahuod ; Alzahrani, Ebraheem O. ; Asiri, Asim ; El-Dessoky, Mohamed M. ; Kuang, Yang. / Tumor growth dynamics with nutrient limitation and cell proliferation time delay. In: Discrete and Continuous Dynamical Systems - Series B. 2017 ; Vol. 22, No. 10. pp. 3771-3782.
@article{4e9e88f95579452fb795590554d8d19e,
title = "Tumor growth dynamics with nutrient limitation and cell proliferation time delay",
abstract = "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.",
keywords = "Nonlinear tumor model, Nutrient limitation, Proliferation, Quiescence, Stability, Time delay",
author = "Ahuod Alsheri and Alzahrani, {Ebraheem O.} and Asim Asiri and El-Dessoky, {Mohamed M.} and Yang Kuang",
year = "2017",
month = "12",
day = "1",
doi = "10.3934/dcdsb.2017189",
language = "English (US)",
volume = "22",
pages = "3771--3782",
journal = "Discrete and Continuous Dynamical Systems - Series B",
issn = "1531-3492",
publisher = "Southwest Missouri State University",
number = "10",

}

TY - JOUR

T1 - Tumor growth dynamics with nutrient limitation and cell proliferation time delay

AU - Alsheri, Ahuod

AU - Alzahrani, Ebraheem O.

AU - Asiri, Asim

AU - El-Dessoky, Mohamed M.

AU - Kuang, Yang

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - Nonlinear tumor model

KW - Nutrient limitation

KW - Proliferation

KW - Quiescence

KW - Stability

KW - Time delay

UR - http://www.scopus.com/inward/record.url?scp=85028696043&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028696043&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2017189

DO - 10.3934/dcdsb.2017189

M3 - Article

AN - SCOPUS:85028696043

VL - 22

SP - 3771

EP - 3782

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 10

ER -