Research paperStability analysis and numerical simulations via fractional calculus for tumor dormancy models
Introduction
Cancer dormancy is a situation in which the number of tumor cells is at a low level during a period of time and may be related to a resistance to therapies, but after this period, cancer cells can resume their growth rapidly [1], [2], [3]. According to Chen et al. [4], there are three major cancer dormancy mechanisms, which are: cellular dormancy, angiogenic dormancy and immunosurveillance. Dormancy is also considered a clinical phenomenon [5] and occurs in many types of cancer, as well as before metastases. Four aspects of cancer dormancy are discussed in [6], pointing out the need for detailed studies on this phenomenon, since recent clinical data diverge on the concepts presented. In terms of mathematical modeling, a model for the interaction dynamics of lymphocyte–tumor cell population is presented in [7], where some solutions are associated with the dormant state. Analyses of possible causative mechanisms of cancer escape from immune–induced dormancy and a review of models related to cancer–immune interactions are presented in [8] and [9], in this order. Some mathematical models associating tumor and antibody interaction with the murine BCL1 lymphoma and how it leads to dormancy of the tumor are presented in [10]. While [11] reviews mathematical modeling of tumor dormancy, focusing on the recurrence of breast cancer, tumor angiogenesis, a description of cellular dormancy and interaction between tumors and the immune system. A cellular automaton model for proliferative growth of solid tumors, including a variety of cell–level tumor–host interactions and different mechanisms for tumor dormancy, such as the effects of the immune system, is presented in [4]. The book [3] presents some research on tumor dormancy from experimental and clinical perspectives, where biological, mathematical and computational models are considered.
The non–integer order calculus, known as fractional calculus (FC), which is the branch of mathematics that deals with the study of integrals and derivatives of non–integer orders, is a very important field in mathematical modeling. Given an ordinary differential equation, to apply fractional modeling, we replace the integer order derivative by a non–integer derivative, usually with an order lower than or equal to the order of the original derivative.
Throughout the few years of this century, many mathematicians and applied researchers have already obtained important results and generalizations in the field of real process modeling from fractional calculus [12], [13], [14], [15], [16], [17], [18], [19].
Although there is no trivial physical, biological or geometrical interpretation for the fractional operators (derivative and integral) [20], [21], fractional order differential equations are naturally related to systems with memory, since the fractional derivatives are usually not local operators, that is, calculating time-fractional derivative at sometime requires all the previous times [22]. Processes with memory exist in several biological systems [23], [24]. Moreover, fractional differential equations can help us to reduce the errors arising from the neglected parameters in modeling real-life phenomena [25], [26], [27].
In engineering, there are also applications of fractional calculus [28], for example, in the analyses of control and dynamical systems [29], [30], [31]. Furthermore, in physics, there are several potential applications of fractional derivatives [32], for example, in the generalization of some classical equations (telegraph, Langenvin) and in quantum mechanics [33], [34], [35], [36].
In medicine, it has been deduced that cell membranes of a biological organism have fractional order electrical conductance and therefore, they are classified in groups of noninteger order models. Fractional derivatives embody essential features of cell rheological behavior and have been successful in the field of rheology [25]. Some mathematical models in HIV and hepatitis show that fractional models are more approximate than their integer order form [25], [37], [38], [39].
Specially in cancer modeling, which is the main focus of this paper, [40] analyzes two immune effectors considering the Caputo fractional derivative, while Dokoumetzidis et al. [41] presents a more theoretical approach on the existence and uniqueness of a cancer model via Caputo–Fabrizio derivative and Yildiz et al. [42] analyzes the behavior of immune and tumor cell populations under the effects of chemo–immunotherapy. A fractional order model of tumor cell growth and their interactions with general immune effector cells are presented in [43], where they numerically solve the equations using the named multi–step generalized differential transform method (MSGDTM), while Baleanu et al. [44] proposes a fractional order model of the cytotoxic T lymphocyte response to a growing tumor cell population and studies the conditions of tumor elimination analytically. A fractional motion diffusion model is applied to pediatric brain tumors in [45].
Motivated by such considerations, the aim of this study is to investigate that scenarios and important changes can be obtained by using fractional calculus in mathematical models about tumor dormancy. The analytical results and numerical simulations are analyzed via comparison with similar works in which the usual calculus was used. Two models from [7], [8] that involve the dynamics between tumor cells and the immune system are considered in this stage. These models have been thoroughly studied as Ordinary Differential Equations (ODE) models.
The paper is organized as follows. Section 2 presents some concepts about stability analysis in the fractional case. The ODE models to be submitted to change in the derivatives, from usual to fractional, are presented in Section 3. Moreover, in this space, important topics such as the balance of units, when necessary, the stability analyses of the equilibrium points, the numerical method used in the computational simulations and the main results obtained are given. Finally, Section 4 presents the concluding remarks.
Section snippets
Fractional operators and stability
The Caputo fractional derivative of order α, denoted by CDα and considered as in [46], [47], is used in this work in some mathematical models and theorems about the stability theory. Its definition can be given by Definition 2.1 The Caputo operator of order α > 0 of a function t > 0, is given bywhere Γ( · ) is the Euler gamma function and m is a natural number, such that .
The fractional operator of Grünwald–Letnikov is very useful in obtaining
Proposed models
Based on [7], [8], in this section two different models describing the same biological phenomenon are presented. The distinct structure of the models allows the construction of new scenarios to investigate the cancer dormancy via fractional calculus in a more in-depth analysis. The Nonstandard Finite Difference (NSFD) method to obtain the numerical solutions for the fractional models are described, aiming at some comparisons to identify which strengths each modeling type presents. In addition,
Conclusions
From two mathematical models already studied with the usual calculus, fractional modeling was the strategy chosen to obtain new scenarios on tumor dormancy induced by the immune system. In the fractional model (6), the use of certain values in the order of the derivative produced qualitative changes in the behavior of some solutions. Otherwise, the values of α in which this occurs represent scenarios where the immune system is successful in maintaining the state of dormancy and has a greater
Competing interests
None.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-profit sectors.
Acknowledgments
The authors would like to thanks the research group CF@FC for the important and productive discussion.
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