Research article

Existence of traveling wave solutions to data-driven glioblastoma multiforme growth models with density-dependent diffusion

  • † Contributed equally as the first author
  • Received: 21 September 2020 Accepted: 15 October 2020 Published: 23 October 2020
  • Mathematical modeling for cancerous disease has attracted increasing attention from the researchers around the world. Being an effective tool, it helps to describe the processes that happen to the tumour as the diverse treatment scenarios. In this paper, a density-dependent reaction-diffusion equation is applied to the most aggressive type of brain cancer, Glioblastoma multiforme. The model contains the terms responsible for the growth, migration and proliferation of the malignant tumour. The traveling wave solution used is justified by stability analysis. Numerical simulation of the model is provided and the results are compared with the experimental data obtained from the reference papers.

    Citation: Ardak Kashkynbayev, Yerlan Amanbek, Bibinur Shupeyeva, Yang Kuang. Existence of traveling wave solutions to data-driven glioblastoma multiforme growth models with density-dependent diffusion[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7234-7247. doi: 10.3934/mbe.2020371

    Related Papers:

  • Mathematical modeling for cancerous disease has attracted increasing attention from the researchers around the world. Being an effective tool, it helps to describe the processes that happen to the tumour as the diverse treatment scenarios. In this paper, a density-dependent reaction-diffusion equation is applied to the most aggressive type of brain cancer, Glioblastoma multiforme. The model contains the terms responsible for the growth, migration and proliferation of the malignant tumour. The traveling wave solution used is justified by stability analysis. Numerical simulation of the model is provided and the results are compared with the experimental data obtained from the reference papers.


    加载中


    [1] Cancer, Fact Sheets, World Health Organization, 2018. Available from: https://www.who.int/en/news-room/fact-sheets/detail/cancer.
    [2] Brain Cancer, Fact Sheets, National Cancer Institute. Available from: https://www.cancer.gov/types/brain/patient/adult-brain-treatment-pdq.
    [3] International Agency for Research on Cancer, Report of World Health Organization, 2018. Available from: http://gco.iarc.fr/today/data/factsheets/populations/900-world-fact-sheets.pdf.
    [4] Brain Tumor: Statistics, eJournal Cancer, Net of American Society of Clinical Oncology, January 2020. Available from: https://www.cancer.net/cancer-types/brain-tumor/statistics.
    [5] Glioblastoma Multiforme: a giving smarter guide to accelerating research progress, Faster Cures Glioblastoma Report, Center of Milken Institute, 2014. Available from: https://www.fastercures.org/assets/Uploads.
    [6] T. Alarcón, H. M. Byrne, P. K. Maini, A multiple scale model for tumor growth, Multiscale Model. Simul., 3 (2005), 440-475. doi: 10.1137/040603760
    [7] K. R. Swanson, E. Alvord, J. Murray, A quantative model for differential motility of gliomas in grey and white matter, Cell Proliferation, 33, (2000), 317-330.
    [8] K. R. Swanson, C. Bridge, J. Murray, E. Alvord, Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216, (2003), 1-10.
    [9] S. E. Eikenberry, T. Sankar, M. Preul, E. Kostelich, C. Thalhauser and Y. Kuang, Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model, Cell Proliferation, 42 (2009), 115-134.
    [10] G. Powathil, M. Kohandel, S. Sivaloganathan, A. Oza, M. Milosevic, Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy, Phys. Med. Biol., 52 (2007), 3291-3306. doi: 10.1088/0031-9155/52/11/023
    [11] H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, K. Swanson, Mathematical modelling of Glioblastoma tumour development: a review, Math. Models Methods Appl. Sci., 24 (2005), 1779-1794.
    [12] N. L. Martirosyan, E. M. Rutter, W. L. Ramey, E. J. Kostelich, Y. Kuang, M. C. Preul, Mathematically modeling the biological properties of Gliomas: a review, Math. Biosci. Eng., 12 (2015), 879-905. doi: 10.3934/mbe.2015.12.879
    [13] J. Murray, Glioblastoma brain tumours: Estimating the time from brain tumour initiation and resolution of a patient survival anomaly after similar treatment protocols, J. Biol. Dyn., 6 (2012), 118-127. doi: 10.1080/17513758.2012.678392
    [14] L. Curtin, A. Hawkins-Daarud, K. G. van der Zee, K. R. Swanson, M. R. Owen, Speed switch in glioblastoma growth rate due to enhanced hypoxia-induced migration, Bull. Math. Biol., 82 (2020), 43.
    [15] P. Gerlee, S. Nelander, The impact of phenotypic switching on glioblastoma growth and invasion, PLoS Comput. Biol., 8, (2012), e1002556.
    [16] T. L. Stepien, E. M. Rutter, Y. Kuang, Traveling waves of a go-or-grow model of glioma growth, SIAM J. Appl. Math., 78, (2018), 1178-1801.
    [17] T. L. Stepien, E. M. Rutter, Y. Kuang, A data-motivated density-dependent diffusion model of in vitro glioblastoma growth, Math. Biosci. Eng., 12, (2015), 1157-1172.
    [18] A. M. Stein, T. Demuth, D. Mobley, M. Berens, L.M. Sander, A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment, Biophys. J., 92, (2007), 356-365.
    [19] H. Enderling, M. A. J. Chaplain, Mathematical modeling of tumor growth and treatment, Curr. Pharm. Des., 20 (2014), 4934-4940. doi: 10.2174/1381612819666131125150434
    [20] Y. Kuang, J. D. Nagy, S. E. Eikenberry, Introduction to mathematical oncology, CRC Press, 2015.
    [21] L. Han, S. Eikenberry, C. He, L. Johnson, M. C. Preul, E. J. Kostelich, Y. Kuang, Patient-specific parameter estimates of glioblastoma multiforme growth dynamics from a model with explicit birth and death rates, Math. Biosci. Eng., 16, (2019), 5307-5323. doi: 10.3934/mbe.2019265
    [22] Z. Wen, M. Fan, A. M. Asiri, E. O. Alzahrani, M. M. El-Dessoky, Y. Kuang, Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model, Math. Biosci. Eng., 14, (2017), 407-420.
    [23] E. M. Rutter, T. L. Stepien, B. J. Anderies, J. D. Plasencia, E. C. Woolf, A. C. Scheck, et al., Mathematical Analysis of Glioma Growth in a Murine Model, Sci. Rep., 7, (2017), 2508.
    [24] R. A. Fisher, The wave of advance of advantageous genes, Ann. Genet., 7 (1937), 355—369.
    [25] A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech., 1 (1937), 1—25.
    [26] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling wave, Memoirs Amer. Math. Soc., 285 (1983), 8—31.
    [27] D. G. Kendall, A form of wave propagation associated with the equation of heat conduction, Proc. Camb. Phil. Soc., 44 (1948), 591-593. doi: 10.1017/S0305004100024609
    [28] D. G. Aronson, H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, in Partial Differential Equations and Related Topics (ed. J.A. Goldstein), Springer, (1975), 5-49.
    [29] D. G. Aronson, H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5
    [30] O. Dieckman, Dynamics in Bio-Mathematical perspective, Math. Comput. Sci. II, 4 (1986), 23-50.
    [31] J. Murray, Mathematical Biology I: An Introduction, 3dedition, Springer, 2002.
    [32] J. Murray, Mathematical Biology II: Spatial models and biomedical applications, 3dedition, Springer, 2003.
    [33] P. Tracqui, G. Cruywagen, D. Woodward, G. Bartoo, J. Murray, E. Alvord, A mathematical model of glioma growth: the effect of extent of surgical resection, Cell Proliferation, 29, (1996), 269-288.
    [34] M. Kot, Elements of mathematical ecology, Cambridge University Press, Cambridge, 2001.
    [35] J. Doke, GRABIT, MATLAB Central File Exchange, 2005. Available from: https: //www.mathworks.com/matlabcentral/fileexchange/7173-grabit.
    [36] J. C. Lagarias, J. A. Reeds, M. H. Wright, E. P. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim., 9 (1998), 112-147. doi: 10.1137/S1052623496303470
    [37] N. J. Armstrong, K. J. Painter, J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theor. Biol., 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3005) PDF downloads(113) Cited by(2)

Article outline

Figures and Tables

Figures(3)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog