Two Numerical Solutions for Solving a Mathematical Model of the Avascular Tumor Growth

AbstractIn this paper, a mathematical model for a solid avascular tumor growth under the effect of periodic therapy is studied. Necessary and sufficient conditions for the global stability of tumor free equilibrium are given. The conditions under which there exists a unique periodic solution to the model are determined and we also show that the unique periodic solution is global attractor of all other positive solutions.

Time delays in proliferation process for solid avascular tumor under the action of external inhibitors

International Journal of Biomathematics ◽

10.1142/s1793524515500187 ◽

2015 ◽

Vol 08 (02) ◽

pp. 1550018

Author(s):

Shihe Xu ◽

Meng Bai

Keyword(s):

Mathematical Model ◽

Immune System ◽

Tumor Growth ◽

Medical Treatment ◽

Time Delays ◽

Dynamical Behavior ◽

Stationary Solutions ◽

The Body ◽

Parameter Values ◽

Avascular Tumor

In this paper a delayed mathematical model for tumor growth under the action of external inhibitors is studied. The delay represents the time taken for cells to undergo mitosis. External inhibitor means that an inhibitor is either developed from the immune system of the body or administered by medical treatment to distinguish with that secreted by tumor itself. Non-negativity of solutions is studied. Local and global stabilities of the stationary solutions are proved for some parameter values. The analysis of the effect of inhibitor's parameters on tumor's growth is presented. The results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding nondelayed model for some parameter values.

A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents

Journal of Theoretical Biology ◽

10.1016/j.jtbi.2006.07.013 ◽

2006 ◽

Vol 243 (4) ◽

pp. 532-541 ◽

Cited By ~ 60

Author(s):

B. Ribba ◽

O. Saut ◽

T. Colin ◽

D. Bresch ◽

E. Grenier ◽

...

Keyword(s):

Mathematical Model ◽

Tumor Growth ◽

Therapeutic Benefit ◽

Multiscale Mathematical Model ◽

Avascular Tumor

A Mathematical Model for the Onset of Avascular Tumor Growth in Response to the Loss of P53 Function

Cancer Informatics ◽

10.1177/117693510600200022 ◽

2006 ◽

Vol 2 ◽

pp. 117693510600200 ◽

Cited By ~ 3

Author(s):

Howard A. Levine ◽

Michael W. Smiley ◽

Anna L. Tucker ◽

Marit Nilsen-Hamilton

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Tumor Growth ◽

Solid Tumor ◽

Tumor Size ◽

P53 Protein ◽

Cell Movement ◽

Inhomogeneous Solution ◽

Cell Expression ◽

Avascular Tumor

We present a mathematical model for the formation of an avascular tumor based on the loss by gene mutation of the tumor suppressor function of p53. The wild type p53 protein regulates apoptosis, cell expression of growth factor and matrix metalloproteinase, which are regulatory functions that many mutant p53 proteins do not possess. The focus is on a description of cell movement as the transport of cell population density rather than as the movement of individual cells. In contrast to earlier works on solid tumor growth, a model is proposed for the initiation of tumor growth. The central idea, taken from the mathematical theory of dynamical systems, is to view the loss of p53 function in a few cells as a small instability in a rest state for an appropriate system of differential equations describing cell movement. This instability is shown (numerically) to lead to a second, spatially inhomogeneous, solution that can be thought of as a solid tumor whose growth is nutrient diffusion limited. In this formulation, one is led to a system of nine partial differential equations. We show computationally that there can be tumor states that coexist with benign states and that are highly unstable in the sense that a slight increase in tumor size results in the tumor occupying the sample region while a slight decrease in tumor size results in its ultimate disappearance.

Analysis of a time-delayed mathematical model for solid avascular tumor growth under the action of external inhibitors

Journal of Applied Mathematics and Computing ◽

10.1007/s12190-015-0947-x ◽

2015 ◽

Vol 52 (1-2) ◽

pp. 403-415 ◽

Cited By ~ 1

Author(s):

Shihe Xu ◽

Yinhui Chen ◽

Meng Bai

Keyword(s):

Mathematical Model ◽

Tumor Growth ◽

Avascular Tumor

Numerical simulations and analysis for mathematical model of avascular tumor growth using Gompertz growth rate function

Alexandria Engineering Journal ◽

10.1016/j.aej.2021.02.040 ◽

2021 ◽

Vol 60 (4) ◽

pp. 3731-3740

Author(s):

Akhtar Ali ◽

Majid Hussain ◽

Abdul Ghaffar ◽

Zafar Ali ◽

Kottakkaran Sooppy Nisar ◽

...

Keyword(s):

Mathematical Model ◽

Growth Rate ◽

Tumor Growth ◽

Numerical Simulations ◽

Rate Function ◽

Growth Rate Function ◽

Avascular Tumor

Deterministic convection-diffusion approach for modeling cell motion and spatial organization: Experimentation on avascular tumor growth

2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) ◽

10.1109/bibm.2017.8217709 ◽

2017 ◽

Author(s):

Cheikhou Oumar Ka ◽

Jean-Marie Dembele ◽

Christophe Cambier ◽

Serge Stinckwich ◽

Moussa Lo ◽

...

Keyword(s):

Tumor Growth ◽

Spatial Organization ◽

Convection Diffusion ◽

Cell Motion ◽

Avascular Tumor

A Meshless Method for Solving the Mathematical Model Associated the Leakage Problem in Gas Pipeline

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.986-987.1418 ◽

2014 ◽

Vol 986-987 ◽

pp. 1418-1421

Author(s):

Jun Shan Li

Keyword(s):

Differential Equation ◽

Mathematical Model ◽

Partial Differential Equation ◽

Inverse Problem ◽

Basis Function ◽

Meshless Method ◽

Numerical Solutions ◽

Basis Functions ◽

Gas Pipeline ◽

The Mathematical Model

In this paper, we propose a meshless method for solving the mathematical model concerning the leakage problem when the pressure is tested in the gas pipeline. The method of radial basis function (RBF) can be used for solving partial differential equation by writing the solution in the form of linear combination of radius basis functions, that is, when integrating the definite conditions, one can find the combination coefficients and then the numerical solution. The leak problem is a kind of inverse problem that is focused by many engineers or mathematical researchers. The strength of the leak can find easily by the additional conditions and the numerical solutions.

A Multiscale Model for Avascular Tumor Growth

Biophysical Journal ◽

10.1529/biophysj.105.060640 ◽

2005 ◽

Vol 89 (6) ◽

pp. 3884-3894 ◽

Cited By ~ 233

Author(s):

Yi Jiang ◽

Jelena Pjesivac-Grbovic ◽

Charles Cantrell ◽

James P. Freyer

Keyword(s):

Tumor Growth ◽

Multiscale Model ◽

Avascular Tumor

A NUMERICAL PROOF THAT CERTAIN CELLS FOLLOW THE TRAILS OF THE DIFFUSIONS OF SOME CHEMICALS IN THE EXTRACELLULAR MATRIX

Journal of Mechanics in Medicine and Biology ◽

10.1142/s0219519421500275 ◽

2021 ◽

pp. 2150027

Author(s):

İREM ÇAY ◽

SERDAL PAMUK

Keyword(s):

Mathematical Model ◽

Extracellular Matrix ◽

Tumor Angiogenesis ◽

Numerical Solutions ◽

Extra Cellular Matrix ◽

The Matrix ◽

Good Agreement ◽

Cellular Matrix

In this work, we obtain the numerical solutions of a 2D mathematical model of tumor angiogenesis originally presented in [Pamuk S, ÇAY İ, Sazci A, A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix, Math Biosci 306:32–48, 2018] to numerically prove that the certain cells, the endothelials (EC), pericytes (PC) and macrophages (MC) follow the trails of the diffusions of some chemicals in the extracellular matrix (ECM) which is, in fact, inhomogeneous. This leads to branching, the sprouting of a new neovessel from an existing vessel. Therefore, anastomosis occurs between these sprouts. In our figures we do see these branching and anastomosis, which show the fact that the cells diffuse according to the structure of the ECM. As a result, one sees that our results are in good agreement with the biological facts about the movements of certain cells in the Matrix.