Mathematical Model of Brain Tumor With Radiotherapy Treatment
DOI:
https://doi.org/10.26713/cma.v14i2.2442Keywords:
Radiotherapy, Malignant glioma cells, Analytical solution, Glial cellsAbstract
A model consisting of three components has been created to describe the interactions among glial cells, glioma cells, and radiotherapy treatment in tumor growth. An analytic solution of nonlinear differential equations is obtained. Stability analysis is discussed under three categories: trivial state, without any treatment, and radiotherapy treatment. In the absence of treatment, the stability analysis of the model demonstrates that a tumor would proliferate to its highest capacity. The treatment of radiotherapy could increase the effectiveness of the fight against gliomas. Moreover, numerical simulations are also provided for the proposed model. Finally, the validity of the system is examined by comparing the graphs of the analytical solution and numerical simulation.
Downloads
References
M. Awadalla, Y. Y. Yameni and K. Abuassba, New Fractional model for the cancer treatment by radiotherapy using the Hadamard fractional derivative, Online Mathematics 1(2) (2019), 14 – 18.
G. Belostotski, A Control Theory Model for Cancer Treatment by Radiotherapy, M.Sc. Thesis, University of Alberta, (2005), DOI: 10.7939/r3-eaqy-7q08.
G. Belostotski and H.I. Freedman, A control theory model for cancer treatment by radiotherapy, International Journal of Pure and Applied Mathematics 25(4) (2005), 447 – 480, URL: http://www.ijpam.eu/contents/2005-25-4/3/3.pdf.
A. Cappuccio, M. A. Herrero and L. Nuñez, Tumor radiotherapy and its mathematical modeling, in: Mathematics, Developmental Biology and Tumour Growth, F. Giráldez and M. A. Herrero (editors), Contemporary Mathematics 402 (2009), 77 – 102, DOI: 10.1090/conm/492/09632.
M. A. Dokuyucu, E. Celik, H. Bulut and H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus 133(2018), Article number: 92, DOI: 10.1140/epjp/i2018-11950-y.
H. I. Freedman and G. Belostotski, Perturbed models for cancer treatment by radiotherapy, Differential Equations and Dynamical Systems 17 (2009), 115 – 133, DOI: 10.1007/s12591-009-0009-7.
K. C. Iarosz, F. S. Borges, A. M. Batista, M. S. Baptista, R. A. N. Siqueira, R. L. Viana and S. R. Lopes, Mathematical model of brain tumour with glia–neuron interactions and chemotherapy treatment, Journal of Theoretical Biology 368(2015), 113-121, DOI: 10.1016/j.jtbi.2015.01.006.
D. J. Kerr, D. G. Haller, C. J. H. van de Velde and M. Baumann, Oxford Textbook of Oncology, 3rd edition, Oxford University Press, (2002), DOI: 10.1093/med/9780199656103.001.0001.
S. Khajanchi and J. J. Nieto, Mathematical modeling of tumor-immune competitive system, considering the role of time delay, Applied Mathematics and Computation 340 (2019), 180 – 205, DOI: 10.1016/j.amc.2018.08.018.
Z. Liu, S. Zhong, C. Yin and W. Chen, Permanence, extinction and periodic solutions in a mathematical model of cell populations affected by periodic radiation, Applied Mathematics Letters 24(10) (2011), 1745 – 1750, DOI: 10.1016/j.aml.2011.04.036.
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Interdisciplinary Applied Mathematics series (IAM), Vol. 18, Springer, New York, NY, xxv + 814 pages (2003), URL: https://link.springer.com/book/10.1007/b98869.
K. W. Okamoto, P. Amarasekare and I. T. D. Petty, Modeling oncolytic virotherapy: Is complete tumor-tropism too much of a good thing?, Journal of Theoretical Biology 358 (2014), 166 – 178,DOI: 10.1016/j.jtbi.2014.04.030.
J. Peiffer and P. Kleihues, Hans-Joachim Scherer (1906-1945), Pioneer in glioma research, Brain Pathology 9(2) (1999), 241 – 245, DOI: 10.1111/j.1750-3639.1999.tb00222.x.
S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade and H. I. Freedman, A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlinear Analysis: Real World Applications 14(1) (2013), 815 – 828, DOI: 10.1016/j.nonrwa.2012.07.034.
R. Rockne, E. C. Alvord Jr., J. K. Rockhill and K. R. Swanson, A mathematical model for brain tumor response to radiation therapy, Journal of Mathematical Biology 58 (2009), 561 – 78, DOI: 10.1007/s00285-008-0219-6.
E. Simbawa, N. Al-Johani and S. Al-Tuwairqi, Modeling the spatiotemporal dynamics of oncolytic viruses and radiotherapy as a treatment for cancer, Computational and Mathematical Methods in Medicine 2020 (2020), Article ID 3642654, 10 pages, DOI: 10.1155/2020/3642654.
J. S. Spratt and T. L. Spratt, Rates of growth of pulmonary metastases and host survival, Annals of Surgery 159(2) (1964), 161 – 171, DOI: 10.1097/00000658-196402000-00001.
V. W. Stieber, Low-grade gliomas, Current Treatment Options in Oncology 2 (2001), 495 – 506, DOI: 10.1007/s11864-001-0071-z.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
