scholarly journals Qualitative and Computational Analysis of a Mathematical Model for Tumor-Immune Interactions

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
F. A. Rihan ◽  
M. Safan ◽  
M. A. Abdeen ◽  
D. Abdel Rahman
Keyword(s):  
Mathematical Model ◽  
Tumor Growth ◽  
Delay Differential Equations ◽  
Computational Analysis ◽  
Tumor Growth Rate ◽  
Cellular Immunotherapy ◽  
Adoptive Cellular Immunotherapy ◽  
Effector Cells ◽  
Two Parameters ◽  
Delay Differential

We provide a family of ordinary and delay differential equations to model the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells and the rate of influx of IL-2. The critical tumor-growth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor dormancy.

scholarly journals MATHEMATICAL MODELING OF TUMOR CELL GROWTH AND IMMUNE SYSTEM INTERACTIONS

2012 ◽  
Vol 09 ◽  
pp. 95-111 ◽  
Author(s):  
FATHALLA A. RIHAN ◽  
MUNTASER SAFAN ◽  
MOHAMED A. ABDEEN ◽  
DUAA H. ABDEL-RAHMAN
Keyword(s):  
Mathematical Modeling ◽  
Tumor Growth ◽  
Delay Differential Equations ◽  
Tumor Growth Rate ◽  
Cellular Immunotherapy ◽  
Tumor Cell Growth ◽  
Adoptive Cellular Immunotherapy ◽  
Effector Cells ◽  
Two Parameters ◽  
Delay Differential

In this paper, we provide a family of ordinary and delay differential equations to describe the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells, and the rate of influx of IL2. The critical tumor-growth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor-dormancy.

Stability of a Mathematical Model with Piecewise Constant Arguments for Tumor-Immune Interaction Under Drug Therapy

2019 ◽  
Vol 29 (01) ◽  
pp. 1950009 ◽  
Author(s):  
Zonghong Feng ◽  
Xinxing Wu ◽  
Luo Yang
Keyword(s):  
Mathematical Model ◽  
Growth Rate ◽  
Drug Therapy ◽  
Tumor Growth ◽  
Tumor Cells ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Tumor Growth Rate ◽  
Chaotic Region ◽  
Treatment Parameter

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].

THREE TYPES OF SIMPLE DDE'S DESCRIBING TUMOR GROWTH

2007 ◽  
Vol 15 (04) ◽  
pp. 453-471 ◽  
Author(s):  
MAREK BODNAR ◽  
URSZULA FORYŚ
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Tumor Growth ◽  
Delay Differential Equations ◽  
Ehrlich Ascites Tumor ◽  
Ascites Tumor ◽  
Ehrlich Ascites ◽  
Delay Differential

In this paper, we compare three types of dynamical systems used to describe tumor growth. These systems are defined as solutions to three delay differential equations: the logistic, the Gompertz and the Greenspan types. We present analysis of these systems and compare with experimental data for Ehrlich Ascites tumor in mice.

A mathematical model of a crocodilian population using delay-differential equations

2008 ◽  
Vol 57 (5) ◽  
pp. 737-754 ◽  
Author(s):  
Angela Gallegos ◽  
Tenecia Plummer ◽  
David Uminsky ◽  
Cinthia Vega ◽  
Clare Wickman ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Delay Differential

scholarly journals Dynamics of competitive interaction between two mining enterprises and one processing: existence and stability of the equilibrium point

2020 ◽  
Vol 193 ◽  
pp. 01051
Author(s):  
Aleksandr Kirjanen ◽  
Oleg Malafeyev ◽  
Irina Zaitseva ◽  
Aleksandr Kovshov ◽  
Dmitry Kolesov
Keyword(s):  
Mathematical Model ◽  
Asymptotic Stability ◽  
Equilibrium Point ◽  
Delay Differential Equations ◽  
Raw Materials ◽  
Dynamic Interaction ◽  
Third Order ◽  
Time Intervals ◽  
Existence And Stability ◽  
Delay Differential

A mathematical model of dynamic interaction between two mining enterprises and one processing is formalized and studied in the paper. The process of interaction is described by a system of three delay differential equations. The criterion for asymptotic stability of nontrivial equilibrium point is obtained when all three enterprises co-work steadily. The problem is reduced to finding stability criterion for quasi-polynomial of third order. We found time intervals between deliveries of raw materials and manufacturing of finished products, in which the process of interaction of the three agents is stable.

scholarly journals In VivoImaging-Based Mathematical Modeling Techniques That Enhance the Understanding of Oncogene Addiction in relation to Tumor Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Chinyere Nwabugwu ◽  
Kavya Rakhra ◽  
Dean Felsher ◽  
David Paik
Keyword(s):  
Mathematical Model ◽  
Tumor Growth ◽  
Cellular Response ◽  
Time Course ◽  
Tumor Regression ◽  
Tumor Burden ◽  
Oncogene Addiction ◽  
Complex Interactions ◽  
Immune Mediated ◽  
Delay Differential

The dependence on the overexpression of a single oncogene constitutes an exploitable weakness for molecular targeted therapy. These drugs can produce dramatic tumor regression by targeting the driving oncogene, but relapse often follows. Understanding the complex interactions of the tumor’s multifaceted response to oncogene inactivation is key to tumor regression. It has become clear that a collection of cellular responses lead to regression and that immune-mediated steps are vital to preventing relapse. Our integrative mathematical model includes a variety of cellular response mechanisms of tumors to oncogene inactivation. It allows for correct predictions of the time course of events following oncogene inactivation and their impact on tumor burden. A number of aspects of our mathematical model have proven to be necessary for recapitulating our experimental results. These include a number of heterogeneous tumor cell states since cells following different cellular programs have vastly different fates. Stochastic transitions between these states are necessary to capture the effect of escape from oncogene addiction (i.e., resistance). Finally, delay differential equations were used to accurately model the tumor growth kinetics that we have observed. We use this to model oncogene addiction in MYC-induced lymphoma, osteosarcoma, and hepatocellular carcinoma.

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