A fractional mathematical model of breast cancer competition model. (English) Zbl 1448.92105
Summary: In this paper, a mathematical model which considers population dynamics among cancer stem cells, tumor cells, healthy cells, the effects of excess estrogen and the body’s natural immune response on the cell populations was considered. Fractional derivatives with power law and exponential decay law in Liouville-Caputo sense were considered. Special solutions using an iterative scheme via Laplace transform were obtained. Furthermore, numerical simulations of the model considering both derivatives were obtained using the Atangana-Toufik numerical method. Also, random model described by a system of random differential equations was presented. The use of fractional derivatives provides more useful information about the complexity of the dynamics of the breast cancer competition model.
MSC:
| 92C50 | Medical applications (general) |
| 92D25 | Population dynamics (general) |
| 34A08 | Fractional ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
Keywords:
fractional derivatives and integrals; integral transforms; Laplace transform; breast cancer competition modelSoftware:
BOADICEAReferences:
| [1] | Siegel, R. L.; Miller, K. D.; Jemal, A., Cancer statistics, 2018, CA Cancer J Clin, 68, 1, 7-30 (2018) |
| [2] | Mufudza, C.; Sorofa, W.; Chiyaka, E. T., Assessing the effects of estrogen on the dynamics of breast cancer, Comput Math Methods Med, 1, 1-14 (2012) · Zbl 1254.92048 |
| [3] | Enderling, H.; Chaplain, M. A.; Anderson, A. R.; Vaidya, J. S., A mathematical model of breast cancer development, local treatment and recurrence, J Theor Biol, 246, 2, 245-259 (2007) · Zbl 1451.92091 |
| [4] | Abernathy, K.; Abernathy, Z.; Baxter, A.; Stevens, M., Global dynamics of a breast cancer competition model, Differ Equations Dyn Syst, 1, 1-15 (2017) |
| [5] | Chen, C.; Baumann, W. T.; Xing, J.; Xu, L.; Clarke, R.; Tyson, J. J., Mathematical models of the transitions between endocrine therapy responsive and resistant states in breast cancer, J R Soc Interface, 11, 96, 1-11 (2014) |
| [6] | Wang, Z.; Butner, J. D.; Kerketta, R.; Cristini, V.; Deisboeck, T. S., Simulating cancer growth with multiscale agent-based modeling, Seminars in cancer biology., vol. 30, 70-78 (2015), Academic Press. |
| [7] | Paine, I.; Chauviere, A.; Landua, J.; Sreekumar, A.; Cristini, V.; Rosen, J.; Lewis, M. T., A geometrically-constrained mathematical model of mammary gland ductal elongation reveals novel cellular dynamics within the terminal end bud, PLoS Comput Biol, 12, 4, 1-23 (2016) |
| [8] | Barrea, A.; Hernández, M. E., Optimal control of a delayed breast cancer stem cells nonlinear model, Optim. Control Appl Methods, 37, 2, 248-258 (2016) · Zbl 1338.49082 |
| [9] | Jenner, A. L.; Yun, C. O.; Kim, P. S.; Coster, A. C., Mathematical modelling of the interaction between cancer cells and an oncolytic virus: insights into the effects of treatment protocols, Bull Math Biol, 1, 1-15 (2018) · Zbl 1396.92032 |
| [10] | Weis, J. A.; Miga, M. I.; Yankeelov, T. E., Three-dimensional image-based mechanical modeling for predicting the response of breast cancer to neoadjuvant therapy, Comput Methods Appl MechEng, 314, 494-512 (2017) · Zbl 1439.92114 |
| [11] | Lee, A. J.; Cunningham, A. P.; Kuchenbaecker, K. B.; Mavaddat, N.; Easton, D. F.; Antoniou, A. C., BOADICEA breast cancer risk prediction model: updates to cancer incidences, tumour pathology and web interface, Br J Cancer, 110, 2, 1-11 (2014) |
| [12] | Owolabi, K. M., Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons Fractals, 93, 89-98 (2016) · Zbl 1372.92091 |
| [13] | Owolabi, K. M.; Atangana, A., Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system, Adv Appl Math Mech, 9, 6, 1438-1460 (2017) · Zbl 1488.65272 |
| [14] | Kumar, S.; Kumar, A.; Baleanu, D., Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger’s equations arise in propagation of shallow water waves, Nonlinear Dyn, 85, 2, 699-715 (2016) · Zbl 1355.76015 |
| [15] | Hristov, J., Electrical circuits of non-integer order: Introduction to an emerging interdisciplinary area with examples, Analysis and simulation of electrical and computer systems., 251-273 (2018), Springer: Springer Cham. |
| [16] | Khader, M.; Kumar, S.; Abbasbandy, S., Fractional homotopy analysis transforms method for solving a fractional heat-like physical model, Walailak J Sci Technol, 13, 5, 337-353 (2015) |
| [17] | Hristov, J., Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models, Frontiers, 1, 270-342 (2017) |
| [18] | Guo, Y. M.; Zhao, Y.; Zhou, Y. M.; Xiao, Z. B.; Yang, X. J., On the local fractional LWR model in fractal traffic flows in the entropy condition, Math Methods Appl Sci, 40, 17, 6127-6132 (2015) · Zbl 1386.90030 |
| [19] | Odibat, Z.; Kumar, S., A robust computational algorithm of homotopy asymptotic method for solving systems of fractional differential equations, J Comput Nonlinear Dyn, 14, 8, 1-8 (2019) |
| [20] | Rahimi, Z.; Sumelka, W.; Yang, X. J., A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams, Eur Phys J Plus, 132, 11, 1-16 (2017) |
| [21] | Kumar, S., A new mathematical model for nonlinear wave in a hyperelastic rod and its analytic approximate solution, Walailak J Sci Technol, 11, 11, 965-973 (2014) |
| [22] | Saad, K. M., An approximate analytical solution of coupled nonlinear fractional diffusion equations, J Fract Calculus Appl, 5, 1, 58-72 (2014) · Zbl 1488.35573 |
| [23] | Du, M.; Wang, Z.; Hu, H., Measuring memory with the order of fractional derivative, Sci Rep, 3, 3431 (2013) |
| [24] | Atangana, A.; Alqahtani, R. T., Tumour model with intrusive morphology, progressive phenotypical heterogeneity and memory, Eur Phys J Plus, 133, 3, 1-10 (2018) |
| [25] | Alkahtani, B. S.T.; Atangana, A.; Koca, I., New nonlinear model of population growth, PloS ONE, 12, 10, 1-12 (2017) |
| [26] | Sardar, T.; Rana, S.; Bhattacharya, S.; Al-Khaled, K.; Chattopadhyay, J., A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector, Math Biosci, 263, 18-36 (2015) · Zbl 1371.92127 |
| [27] | Owolabi, K. M., Numerical solution of diffusive HBV model in a fractional medium, SpringerPlus, 5, 1, 1-19 (2016) |
| [28] | Koca, I., Analysis of rubella disease model with non-local and non-singular fractional derivatives, Int J OptimControl, 8, 1, 17-25 (2018) |
| [29] | Caputo, M.; Mainardi, F., A new dissipation model based on memory mechanism, Pure Appl Geophys, 91, 134-147 (1971) · Zbl 1213.74005 |
| [30] | Toufik, M.; Atangana, A., New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur Phys J Plus, 132, 10, 1-16 (2017) |
| [31] | Caputo, M.; Fabricio, M., A new definition of fractional derivative without singular Kernel, Progr Fract Differ Appl, 1, 2, 73-85 (2015) |
| [32] | Lozada, J.; Nieto, J. J., Properties of a new fractional derivative without singular Kernel, Progr Fract Differ Appl, 1, 2, 87-92 (2015) |
| [33] | Atangana, A.; Jain, S., The role of power decay, exponential decay and Mittag-Leffler function’s waiting time distribution: application of cancer spread, Physica A, 512, 330-351 (2018) · Zbl 1514.65084 |
| [34] | Misirli, E.; Gurefe, Y., Multiplicative Adams Bashforth-Moulton methods, Numer Algorithms, 57, 4, 425-439 (2011) · Zbl 1221.65167 |
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