Modeling cancer evolution: evolutionary escape under immune system control

Korobeinikov, Andrei; Starkov, Konstantin E.; Valle, Paul A.

doi:10.1088/1742-6596/811/1/012004

Cited by 13 publications

(8 citation statements)

References 10 publications

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“…In real life, we can consider one cancer cell as a finite critical value below which tumor survival is impossible. Then, if there is a solution such that ( ) goes below this threshold, one should conclude the complete elimination of the cancer cells population; see the works of d'Onofrio [40] and Korobeinikov [30].…”

Section: Discussionmentioning

confidence: 99%

“…About the different dynamics of the system, numerical simulations illustrate the following concerning the values shown in Table 1: when 12 = 1, the system exhibits chaotic behavior; if 12 < 1, the trajectories of the system will begin to show oscillations such as stable limit cycles and periodic orbits, whereas a value 12 > 1 implies that all solutions will converge to a healthy tumorfree equilibrium point. Although the system exhibits a wide variety of biologically meaningful dynamics, it is important to notice that it does not take into account the evolutionary evasion of tumors from the immune system which is a very important phenomenon that should be considered in the modelling of a disease as complex as cancer [27][28][29][30].…”

Section: The Chaotic-cancer Mathematical Modelmentioning

confidence: 99%

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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: Discussionmentioning

confidence: 99%

Section: The Chaotic-cancer Mathematical Modelmentioning

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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“…We develop a model by assuming the logistic growth of cell populations in the absence of chemotherapy and radiotherapy. Some tumor cells are assumed to avoid immune response control due to succession of mutations leading to the development of immune-resistant cells [27]. At any time t , we consider immune response as natural killer cells denoted by ( I ( t )) and describe its dynamics by assuming that the source of I ( t ) is constantly infused in the body daily.…”

Section: Methodsmentioning

confidence: 99%

“…There is a lot of literature that addresses the development of various mathematical models of cancer and treatment, for example, [16–26]. The work in [27] demonstrated the crucial role played by the immune system in the process of tumor elimination. However, the results showed that despite immune pressure, cancer is able to persist if the cells are able to mutate fast and the immune response is not strong enough.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Breast Cancer under Different Rates of Chemoradiotherapy

Mkango

Shaban

Mureithi

et al. 2019

Computational and Mathematical Methods in Medicine

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A type of cancer which originates from the breast tissue is referred to as breast cancer. Globally, it is the most common cause of death in women. Treatments such as radiotherapy, chemotherapy, hormone therapy, immunotherapy, and gene therapy are the main strategies in the fight against breast cancer. The present study aims at investigating the effects of the combined radiotherapy and chemotherapy as a way to treat breast cancer, and different treatment approaches are incorporated into the model. Also, the model is fitted to data on patients with breast cancer in Tanzania. We determine new treatment strategies, and finally, we show that when sufficient amount of chemotherapy and radiotherapy with a low decay rate is used, the drug will be significantly more effective in combating the disease while health cells remain above the threshold.

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“…Models of immune system-cancer interactions have incorporated a wide range of mathematical approaches, including differential equations, cellular automata, and spatial and non-spatial multiscale models [173][174][175]. These models have examined tumour progression in the immune context [176][177][178], cancer cell-immune cell interactions in general [179,180], and immune suppression and escape [181]. Furthermore, some have incorporated microenvironmental or stromal elements such as paracrine signalling [182], stiffness sensing [183], or considered cancer-immune interactions as they effect therapy response [184,185].…”

Section: Outlook and Concluding Remarksmentioning

confidence: 99%

Dynamic interplay between tumour, stroma and immune system can drive or prevent tumour progression

Seager

Hajal

Spill

et al. 2017

Converg. Sci. Phys. Oncol.

135

102

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In the tumour microenvironment, cancer cells directly interact with both the immune system and the stroma. It is firmly established that the immune system, historically believed to be a major part of the body’s defence against tumour progression, can be reprogrammed by tumour cells to be ineffective, inactivated, or even acquire tumour promoting phenotypes. Likewise, stromal cells and extracellular matrix can also have pro-and anti-tumour properties. However, there is strong evidence that the stroma and immune system also directly interact, therefore creating a tripartite interaction that exists between cancer cells, immune cells and tumour stroma. This interaction contributes to the maintenance of a chronically inflamed tumour microenvironment with pro-tumorigenic immune phenotypes and facilitated metastatic dissemination. A comprehensive understanding of cancer in the context of dynamical interactions of the immune system and the tumour stroma is therefore required to truly understand the progression toward and past malignancy.

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Modeling cancer evolution: evolutionary escape under immune system control

Cited by 13 publications

References 10 publications

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Dynamics of Breast Cancer under Different Rates of Chemoradiotherapy

Dynamic interplay between tumour, stroma and immune system can drive or prevent tumour progression

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