On the global dynamics of a chronic myelogenous leukemia model

Крищенко, А. П.; Starkov, Konstantin E.

doi:10.1016/j.cnsns.2015.10.001

Cited by 29 publications

(16 citation statements)

References 16 publications

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“…The analysis of the dynamics of a superficial bladder cancer model with Bacillus Calmette-Guérin immunotherapy is shown in [24], in which sufficient conditions of global asymptotic stability of the tumor-free equilibrium point are derived by complementing the analysis with the LCIS method. Sufficient conditions to prove that dynamics of a three-dimensional chronic myelogenous leukemia model around the tumor-free equilibrium point is unstable are presented in [25]; further, ultimate upper bounds are computed for all three cell populations and existence conditions of a positively invariant polytope are provided. In [19] authors study the global dynamics of a cancer chemotherapy system and determine sufficient conditions for tumor clearance and global asymptotic stability of the tumorfree equilibrium point by means of the LCIS method and LaSalle's invariance principle.…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Valle

Coria

Gamboa

et al. 2018

Mathematical Problems in Engineering

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The complexity of cancer has motivated the development of different approaches to understand the dynamics of this large group of diseases. One that may allow us to better comprehend the behavior of cancer cells, in both short- and long-term, is mathematical modelling through ordinary differential equations. Several ODE mathematical models concerning tumor evolution and immune response have been formulated through the years, but only a few may exhibit chaotic attractors and oscillations such as stable limit cycles and periodic orbits; these dynamics are not that common among cancer systems. In this paper, we apply the Localization of Compact Invariant Sets (LCIS) method and Lyapunov stability theory to investigate the global dynamics and the main factors involved in tumor growth and immune response for a chaotic-cancer system presented by Itik and Banks in 2010. The LCIS method allows us to compute what we define as the localizing domain, which is formulated by the intersection of all lower and upper bounds of each cells population in the nonnegative octant, R+,03. Bounds of this domain are given by inequalities in terms of the system parameters. Then, we apply Lyapunov stability theory and LaSalle’s invariance principle to establish existence conditions of a global attractor. The latter implies that given any nonnegative initial condition, all trajectories will go to the largest compact invariant set (a stable equilibrium point, limit cycles, periodic orbits, or a chaotic attractor) located either inside or at the boundaries of the localizing domain. In order to complement our analysis, numerical simulations are performed throughout the paper to illustrate all mathematical results and to better explain their biological implications.

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Section: Mathematical Preliminariesmentioning

confidence: 99%

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Valle

Coria

Gamboa

et al. 2018

Mathematical Problems in Engineering

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“…Moreover, we establish the following theorem, which can be found in reference [22], and will be useful in the proof of lemma 3.3. Figure 1.…”

Section: Thusmentioning

confidence: 99%

Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy

Álvarez-Arenas¹,

Starkov²,

Calvo³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, a non-trivial generalization of a mathematical model put forward in [35] to account for the development of resistance by tumors to chemotherapy is presented. A study of the existence and local stability of the solutions, as well as the ultimate dynamics of the model, is addressed. An analysis of different chemotherapeutical protocols using discretization and optimization methods is carried out. A number of objective functionals are considered and the necessary optimality conditions are provided. Since the control variable appears linearly in the associated problem, optimal controls are concatenations of bang-bang and singular arcs. A formula of the singular control in terms of state and adjoint variables is derived analytically. Bangbang and singular controls from the numerical simulations are obtained where, in particular, singular controls illustrate the metronomic chemotherapy.

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“…A sampling based analysis led to the model prediction in which the Gompertzian growth rate as well as the natural death rate of CML cells (without altering the other model parameters) are the key parameters for the control of CML progression. The CML model due to Moore and Li [28] is revisited in [29] and a rigorous analysis of the global dynamics for the same is carried out. The authors established that the dynamics of the system is unstable around the tumor-free equilibrium and obtained the global stability conditions for the internal tumor equilibrium.…”

Section: Introductionmentioning

confidence: 99%

A quantitative study on the role of TKI combined with Wnt/β-catenin signaling and IFN-α in the treatment of CML through deterministic and stochastic approaches

Pan

Raha

Chakrabarty

2020

Chaos, Solitons & Fractals

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We propose deterministic and stochastic models for studying the pharmacokinetics of chronic myeloid leukemia (CML), upon administration of IFN-α (the traditional treatment for CML), TKI (the current frontline medication for CML) and Wnt/β-catenin signaling (the state-of-the art therapeutic breakthrough for CML). To the best of our knowledge, no mathematical model incorporating all these three therapeutic protocols are available in literature. Further, this work introduces a stochastic approach in the study of CML dynamics.The key contributions of this work are: (1) Determination of the patient condition, contingent upon the patient specific model parameters, which leads to prediction of the appropriate patient specific therapeutic dosage. (2) Addressing the question of how the dual therapy of TKI and Wnt/β-catenin signaling or triple combination of all three, offers potentially improved therapeutic responses, particularly in terms of reduced side effects of TKI or IFN-α. (3) Prediction of the likelihood of CML extinction/remission based on the level of CML stem cells at detection.

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On the global dynamics of a chronic myelogenous leukemia model

Cited by 29 publications

References 16 publications

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy

A quantitative study on the role of TKI combined with Wnt/β-catenin signaling and IFN-α in the treatment of CML through deterministic and stochastic approaches

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