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

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value [Formula: see text] is predicted of the treatment parameter [Formula: see text]. We can see Neimark–Sacker bifurcation of the system when [Formula: see text], and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when [Formula: see text].

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Formulation and numerical simulations of a continuum model of avascular tumor growth

Mathematical Biosciences ◽

10.1016/j.mbs.2011.03.002 ◽

2011 ◽

Vol 231 (2) ◽

pp. 159-171 ◽

Cited By ~ 8

Author(s):

Mohammed Shuker Mahmood ◽

Silvia Mahmood ◽

Dušan Dobrota

Keyword(s):

Tumor Growth ◽

Numerical Simulations ◽

Continuum Model ◽

Avascular Tumor

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Logistic equation with thep-Laplacian and constant yield harvesting

Abstract and Applied Analysis ◽

10.1155/s1085337504311097 ◽

2004 ◽

Vol 2004 (9) ◽

pp. 723-727 ◽

Cited By ~ 26

Author(s):

Shobha Oruganti ◽

Junping Shi ◽

Ratnasingham Shivaji

Keyword(s):

Growth Rate ◽

Elliptic Equation ◽

Positive Constant ◽

Quasilinear Elliptic Equation ◽

Rate Function ◽

Logistic Equation ◽

Solution Methods ◽

Growth Rate Function ◽

Quasilinear Elliptic ◽

Constant Yield

We consider the positive solutions of a quasilinear elliptic equation withp-Laplacian, logistic-type growth rate function, and a constant yield harvesting. We use sub-super-solution methods to prove the existence of a maximal positive solution when the harvesting rate is under a certain positive constant.

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Qualitative analysis of a mathematical model for tumor growth under the effect of periodic therapy

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2014-0030 ◽

2015 ◽

Vol 0 (0) ◽

Cited By ~ 1

Author(s):

Shihe Xu

Keyword(s):

Mathematical Model ◽

Periodic Solution ◽

Tumor Growth ◽

Qualitative Analysis ◽

Global Stability ◽

Global Attractor ◽

Positive Solutions ◽

Sufficient Conditions ◽

Necessary And Sufficient ◽

Avascular Tumor

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.

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Growth-rate function for the nonlinear analysis of the transient dynamics of microwave oscillators

2016 IEEE MTT-S International Microwave Symposium (IMS) ◽

10.1109/mwsym.2016.7540302 ◽

2016 ◽

Cited By ~ 3

Author(s):

Almudena Suarez ◽

Sergio Sancho ◽

Franco Ramirez

Keyword(s):

Growth Rate ◽

Nonlinear Analysis ◽

Rate Function ◽

Transient Dynamics ◽

Microwave Oscillators ◽

Growth Rate Function

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Optimal Periodic Harvesting Strategy for the General Autonomous System of Single Specie

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.518-523.237 ◽

2012 ◽

Vol 518-523 ◽

pp. 237-240 ◽

Cited By ~ 1

Author(s):

Zai Le He ◽

Xiang Qing Zhao ◽

Hai Ling Lu

Keyword(s):

Optimal Control ◽

Growth Rate ◽

Optimal Control Problem ◽

Control Problem ◽

Optimal Trajectory ◽

Autonomous System ◽

Rate Function ◽

Growth Rate Function

We study a optimal control problem-optimal periodic harvesting strategy for the general autonomous system of single specie. Suppose the growth rate function f(x) is C^2, x* satisfies f'(x*)=0. We proved that if f"(x*)<0，then x* is the optimal trajectory, otherwise, x* is not the optimal trajectory.

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A Novel Mathematical Model of Tumor Growth Kinetics with Allee Effect under Fuzzy Environment

10.21203/rs.3.rs-730827/v1 ◽

2021 ◽

Author(s):

Rubeena Khaliq ◽

Pervaiz Iqbal ◽

Shahid Ahmad Bhat

Keyword(s):

Mathematical Model ◽

Growth Rate ◽

Tumor Growth ◽

Cell Population ◽

Growth Kinetics ◽

Allee Effect ◽

Growth Patterns ◽

Initial Population ◽

Fuzzy Environment ◽

Tumor Growth Kinetics

Abstract Most of the cancer growth models have described the exponential growth patterns at the very initial stage with low cell population density. Eventually, decreasing the tumor growth rate at higher cell population densities because of deficiency in resources such as space and nutrients. However, recent studies at clinical and preclinical investigations of cancer initiation or reappaearance showed a population dynamics evincing that the growth rate increases as cell number increases. Hence, showing behaviour analogous to cooperative mechanism in the ecosystem and ecological effect called Allee effect. Based on these observations with two arguments i.e. change in initial population and growth rate. In this paper, the novel mathematical model of tumor growth kinetics with Allee effect under fuzzy environment is proposed. In this model the Generalized Hukuhara derivative approach is utilized to solve the fuzzy differential equations. Moreover, it is showen that the change in initial value and growth rate affects the cell density with the Allee effect under the fuzzy environment. Finally the superiority of model has been showen with the help of numerical simulation.

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Disturbance Attenuation via Output Feedback for Uncertain Nonlinear Systems with Output and Input Depending Growth Rate

Mathematical Problems in Engineering ◽

10.1155/2014/621783 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Weiyong Yu ◽

Fangfang Zhang

Keyword(s):

Growth Rate ◽

Nonlinear Systems ◽

Output Feedback ◽

Closed Loop ◽

Rate Function ◽

Disturbance Attenuation ◽

Uncertain Nonlinear Systems ◽

Closed Loop System ◽

Growth Rate Function ◽

Dynamic Gain

The problem of output feedback disturbance attenuation is investigated for a class of uncertain nonlinear systems. The uncertainties of the considered systems are bounded by unmeasured states with growth rate function of output and input multiplying an unknown constant. Based on a dynamic gain observer, an adaptive output feedback controller is proposed such that the states of the closed-loop system are globally bounded, and the disturbance attenuation is achieved in theL2-gain sense. An example is provided to demonstrate the effectiveness of the proposed design scheme.

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Finite Element Modeling of Avascular Tumor Growth Using a Stress-Driven Model

Journal of Biomechanical Engineering ◽

10.1115/1.4037038 ◽

2017 ◽

Vol 139 (8) ◽

Cited By ~ 3

Author(s):

Faezeh Iranmanesh ◽

Mohammad Ali Nazari

Keyword(s):

Growth Rate ◽

Finite Element ◽

Tumor Growth ◽

Finite Element Modeling ◽

Solid Phase ◽

Collagen Fibers ◽

Tumor Growth Rate ◽

Fiber Distribution ◽

Element Modeling ◽

Avascular Tumor

Tumor growth being a multistage process has been investigated from different aspects. In the present study, an attempt is made to represent a constitutive-structure-based model of avascular tumor growth in which the effects of tensile stresses caused by collagen fibers are considered. Collagen fibers as a source of anisotropy in the structure of tissue are taken into account using a continuous fiber distribution formulation. To this end, a finite element modeling is implemented in which a neo-Hookean hyperelastic material is assigned to the tumor and its surrounding host. The tumor is supplied with a growth term. The growth term includes the effect of parameters such as nutrient concentration on the tumor growth and the tumor's solid phase content in the formulation. Results of the study revealed that decrease of solid phase is indicative of decrease in growth rate and the final steady-state value of tumor's radius. Moreover, fiber distribution affects the final shape of the tumor, and it could be used to control the shape and geometry of the tumor in complex morphologies. Finally, the findings demonstrated that the exerted stresses on the tumor increase as time passes. Compression of tumor cells leads to the reduction of tumor growth rate until it gradually reaches an equilibrium radius. This finding is in accordance with experimental data. Hence, this formulation can be deployed to evaluate both the residual stresses induced by growth and the mechanical interactions with the host tissue.

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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.

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