Optimal Control of a Cancer Cell Model with Delay

Collins, Charles T.; Fister, K. Renee; Williams, Molly

doi:10.1051/mmnp/20105305

Cited by 11 publications

(3 citation statements)

References 15 publications

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“…Delay differential equations are widely used for tumor growth modelling, as the introduction of time delays in an ordinary tumor growth model significantly describes the complex cellular interaction and gives us a more realistic picture of the model [21][22][23][24][25][26][27][28]. The inclusion of time delays in tumor immune interaction also produces the nonlinear behavior of tumors [29].…”

Section: Introductionmentioning

confidence: 99%

A study on the dynamics of a breast cancer model with discrete-time delay

Das,

Dehingia,

Hinçal

et al. 2024

Phys. Scr.

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This study aims to discuss the impact of discrete-time delay on the anti-tumor immune response against tumor growth, excess levels of estrogen, and the source rate of immune cells in a breast cancer model. The non-negativity and boundedness of the solutions of the model are discussed. The existence of equilibria and their stability are examined. It is found that if the estrogen level is normal and the source rate of immune cells is low, the stability of the model around the co-existing equilibrium switches to instability via a Hopf bifurcation as the time delay increases. To validate the theoretical findings, a few numerical examples have been presented. The main result of this study is that the growth of tumors can be controlled if the immune system quickly generates an anti-tumor immune response. However, if the immune system takes a longer time to generate anti-tumor immune responses, the tumor growth cannot be controlled, and the system becomes unstable, which may result in the further spread of the disease.

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Section: Introductionmentioning

confidence: 99%

A study on the dynamics of a breast cancer model with discrete-time delay

Das,

Dehingia,

Hinçal

et al. 2024

Phys. Scr.

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“…In literature, applications of optimal control theory to mathematical models of cancer biology and role of chemotherapy began to appear in the 1980s and have appeared with regularity in the following years to the present day. The reader can refer to some of these works, such as for example (Swan, 1980 , 1988 , 1990 ; Martin, 1992 ; Swierniak et al, 1996 ; Kimmel and Swierniak, 2005 ; Pillis et al, 2008 ; Collins et al, 2010 ; Batmani and Khaloozadeh, 2011 ; Ledzewicz et al, 2013 ; Ghaffari et al, 2014 ; Michor and Beal, 2015 ; Wang and Schättler, 2016 ; Carrère, 2017 ; Irurzun-Arana et al, 2018 ). Indeed, a tumor undergoing pharmacological treatment can be viewed as a control system with the state of the system, given by the number of cancer cells at time t , N ( t ), and the control input at time t , u ( t ).…”

Section: Introductionmentioning

confidence: 99%

Control Theory and Cancer Chemotherapy: How They Interact

Lecca

2021

Front. Bioeng. Biotechnol.

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Control theory arises in most modern real-life applications, not least in biological and medical applications. In particular, in biological and medical contexts, the role of control theory began to take shape in the early 1980s when the first works appeared on the application of control theory in models of pharmacokinetics and pharmacodynamics for antitumor therapies. Forty years after those first works, the theory of control continues to be considered a mathematical analysis tool of extreme importance and usefulness, but the challenges it must overcome in order to manage the complexity of biological processes are in fact not yet overcome. In this article, we introduce the reader to the basic ideas of control theory, its aims and its mathematical formalization, and we review its use in cell phase-specific models for cancer chemotherapy. We discuss strengths and limitations of the control theory approach to the analysis pharmacokinetics and pharmacodynamics models, and we will see that most of them are strongly related to data availability and mathematical form of the model. We propose some future research directions that could prove useful in overcoming the these limitations and we indicate the crucial steps preliminary to a useful and informative application of control theory to cancer chemotherapy modeling.

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“…As we have proven the existence of the solution to the system given the control and developed the optimality conditions in [4], we reiterate the conditions here and move toward the discussion of the numerical situations.…”

Section: Objective Functionalmentioning

confidence: 99%

Delay Dynamics of Cancer and Immune Cell Model

Adongo

Fister

2012

Math. Model. Nat. Phenom.

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Abstract. We investigate optimal control of a cancer-immune cell interactive model with delay in the interphase compartment. By applying the optimal control theory, we seek to minimize the cost associated with the chemotherapy drug, minimize the accumulation of cancer cells, and increase the immune cell presence. Optimality conditions and characterization of the control are provided. Numerical analyses are given to enhance the understanding of the difficulties that occur in the control of cancer.

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Optimal Control of a Cancer Cell Model with Delay

Cited by 11 publications

References 15 publications

A study on the dynamics of a breast cancer model with discrete-time delay

A study on the dynamics of a breast cancer model with discrete-time delay

Control Theory and Cancer Chemotherapy: How They Interact

Delay Dynamics of Cancer and Immune Cell Model

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