Research article Special Issues

Mathematical modeling and numerical simulation of a multiscale cancer invasion of host tissue

  • Received: 27 December 2019 Accepted: 02 March 2020 Published: 24 March 2020
  • MSC : 35K57, 92C17

  • In this study, we develop an advection-reaction-diffusion system of partial differential equations (PDEs) to describe interactions between tumor cells and extracellular matrix (ECM) at the macroscopic level. At the subcellular level, we model the interaction of proteolytic enzymes and the ECM with a set of ordinary differential equations (ODEs). A contractivity function is used to couple the macroscopic and microscopic events. The model is supplemented with nutrients supply from the underlying tissue. These PDE-ODE systems of equations model the on-set of tumor cell invasion of the host extracellular matrix. The model accounts for different time and spatial scales at the macroscopic and microscopic levels. Contact inhibition between the tumor cells and the tumor micro-environment are also accounted for through a nonlinear density-dependent diffusion and haptotaxis coefficients. In the numerical simulations, we use a nonstandard finite difference method to illustrate the model predictions. Qualitatively, our results confirm the three distinct layers of proliferating, quiescent and necrotic cells as observed in multicellular spheroids experiments.

    Citation: Peter Romeo Nyarko, Martin Anokye. Mathematical modeling and numerical simulation of a multiscale cancer invasion of host tissue[J]. AIMS Mathematics, 2020, 5(4): 3111-3124. doi: 10.3934/math.2020200

    Related Papers:

  • In this study, we develop an advection-reaction-diffusion system of partial differential equations (PDEs) to describe interactions between tumor cells and extracellular matrix (ECM) at the macroscopic level. At the subcellular level, we model the interaction of proteolytic enzymes and the ECM with a set of ordinary differential equations (ODEs). A contractivity function is used to couple the macroscopic and microscopic events. The model is supplemented with nutrients supply from the underlying tissue. These PDE-ODE systems of equations model the on-set of tumor cell invasion of the host extracellular matrix. The model accounts for different time and spatial scales at the macroscopic and microscopic levels. Contact inhibition between the tumor cells and the tumor micro-environment are also accounted for through a nonlinear density-dependent diffusion and haptotaxis coefficients. In the numerical simulations, we use a nonstandard finite difference method to illustrate the model predictions. Qualitatively, our results confirm the three distinct layers of proliferating, quiescent and necrotic cells as observed in multicellular spheroids experiments.


    加载中


    [1] H. Jan, R. Kathy, W. Wendy, et al. A guide to cancer drug development and regulation, Astra Zeneca, 2006.
    [2] F. Cornelis, O. Saut, P. Comsille, et al. In vivo mathematical modeling of tumor growth from imaging data: Soon to come in the future? Diagn. Interv. Imag., 94 (2013), 593-600.
    [3] G. Meral, C. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete and continuous dynamical systems series B, 20 (2014), 189-213. doi: 10.3934/dcdsb.2015.20.189
    [4] J. Kelkel and C. Surulescu, On some models for cancer cell migration through tissue networks, Math. Biosci. Eng., 8 (2011), 575-589. doi: 10.3934/mbe.2011.8.575
    [5] B. Yue, Biology of the Extracellular Matrix: An Overview, J. Glaucoma, 23 (2014), 20-23. doi: 10.1097/IJG.0000000000000108
    [6] M. A. J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399
    [7] N. Kolbem, A tumor invasion model for heterogeneous cancer cell populations: mathematical analysis and numerical methods, Ph.D thesis, der Johannes Gutenberg-Universität Mainz, 2017. Available from: https://https://publications.ub.uni-mainz.de/theses/volltexte/2018/100001842/pdf/100001842.pdf.
    [8] N. Sfakianakis, N. Kolbe, N. Hellmann, et al. 2016. A multiscale approach to the migration of cancer stem cells: mathematical modelling and simulations, B. Math. Biol., 79 (2017), 209-235. doi: 10.1007/s11538-016-0233-6
    [9] P. Domschke, D. Trucu, A. Gerisch, et al. Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumor infinitive growth patterns, J. Theor. Biol., 361 (2014), 41-60. doi: 10.1016/j.jtbi.2014.07.010
    [10] V. Andasari, A. Gerisch, G. Lolas, et al. Mathematical modelling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1
    [11] P. Lu, T. Dumitru, L. Ping, et al. Multiscale Mathematical Model of Tumour Invasive Growth, B. Math. Biol., 79 (2017), 389-429. doi: 10.1007/s11538-016-0237-2
    [12] T. Dumitru, Multiscale Modelling of Cancer Invasion, 6th European Conference on Computational Mechanics (ECCM 6), 2018.
    [13] R. M. Sutherland, Cell and environment interaction in tumour microregions: the multicell spheroid model, Science, 240 (1988), 177-184. doi: 10.1126/science.2451290
    [14] N. Kolbe, M. Lukacova-Medvid ova, N. Sfakianakis, et al. Numerical simulation of a contractivity based multiscale cancer invasion model, Institute of Math., Johannes Gutenberg-Uni., Mainz, Germany, 2016.
    [15] J. A. Sherratt, M. J. A. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312. doi: 10.1007/s002850100088
    [16] Y. Tao, C. Cui, A density-dependent chemotaxis-haptotaxis system modeling cancer invasion, J. Math. Anal. Appl., 367 (2010), 612-624. doi: 10.1016/j.jmaa.2010.02.015
    [17] R. A. Gatenby, E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753.
    [18] P. Domschke, D. Trucu, A. Gerisch, et al. Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns, J. Theor. Biol., 361 (2014), 40-61. doi: 10.1016/j.jtbi.2014.07.010
    [19] A. Zhigun, C. Surulescu, A. Hunt, Global existence for a degenerate haptotaxis model of tumor invasion under the go-or-grow dichotomy hypothesis, preprint.
    [20] W. Mueller-Klieser, Method for determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids, Biophys. J., 46 (1984), 343-348. doi: 10.1016/S0006-3495(84)84030-8
    [21] W. Mueller-Klieser, J. P. Freyer, R. M. Sutherland, Influence of glucose and oxygen supply conditions on the oxygenation of multicellular spheroids, Brit. J. Cancer, 53 (1986), 345-353. doi: 10.1038/bjc.1986.58
    [22] T. Roose, S. J. Chapman, P. K. Maini, Mathematical Models of Avascular Tumor Growth, Siam rev., 49 (2007), 179-208. doi: 10.1137/S0036144504446291
    [23] T. S. Deisboeck, Z. Wang, P. Macklin, et al. Multiscale Cancer Modeling, Annual review of biomedical engineering, 2010.
    [24] D. Hanahan and R. A. Weinberg, The Hallmarks of Cancer, Cell, 100 (2000), 57-70. doi: 10.1016/S0092-8674(00)81683-9
    [25] D. Hanahan, R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013
    [26] B. Qian, J. W. Pollard, Macrophage Diversity Enhances Tumor Progression and Metastasis, Cell, 141 (2010), 39-51. doi: 10.1016/j.cell.2010.03.014
    [27] J. A. Joyce, J. W. Pollard, Microenvironmental regulation of metastasis, Nat. Rev. Cancer, 9 (2009), 239-252. doi: 10.1038/nrc2618
    [28] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. S., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947
    [29] V. Andasari, R. T. Roper, M. H. Swat, et al. Integrating Intracellular Dynamics Using CompuCell3D and Bionetsolver: Applications to Multiscale Modelling of Cancer Cell Growth and Invasion, PLoS ONE, 7 (2012), e33726. doi: 10.1371/journal.pone.0033726
    [30] M. A. J. Chaplain and N. Deakin, Mathematical modeling of cancer invasion: the role of membrane-bound matrix metalloproteinases, Frontiers in oncology, 2013.
    [31] C. Walker, E. Mojares, A. D. R. Hernández, Role of Extracellular Matrix in Development and Cancer Progression, Int. J. Mol. Sci., 19 (2018), 3028.
    [32] Y. T. N. Edalgo, A. N. F. Versypt, Mathematical Modeling of Metastatic Cancer Migration through a Remodeling Extracellular Matrix, Processes, 6 (2018), 58.
    [33] P. J. Murray, C. M. Edwards, M. J. Tindall, et al. From a discrete to a continuum model of cell dynamics in one dimension, Phys. Rev. E, 80 (2009), 31912. doi: 10.1103/PhysRevE.80.031912
    [34] A. Gerischa, M. A. J. Chaplain, Robust numerical methods for taxis-diffusion-reaction systems: Applications to biomedical problems, Math. Comput. Model., 43 (2006), 49-75. doi: 10.1016/j.mcm.2004.05.016
  • 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(5223) PDF downloads(641) Cited by(5)

Article outline

Figures and Tables

Figures(3)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog