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Networks and Heterogeneous Media

2008, Volume 3, Issue 1: 43-83. doi: 10.3934/nhm.2008.3.43
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Multiphase modeling and qualitative analysis of the growth of tumor cords

  • 1.
    Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino
  • Received: 01 May 2007 Revised: 01 January 2008

  • Primary: 35R35; Secondary: 92B05, 92C37.

  • In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth of the tumor mass. By weakening the role of the extracellular matrix, which is regarded as a rigid non-remodeling scaffold, a system of two partial differential equations is derived, describing the evolution of the cell volume ratio coupled to the dynamics of the nutrient, whose higher and lower concentration levels determine proliferation or death of tumor cells, respectively. Numerical simulations of a reference two-dimensional problem are shown and commented, and a qualitative mathematical analysis of some of its key issues is proposed.

    Citation: Andrea Tosin. Multiphase modeling and qualitative analysis of the growth of tumor cords[J]. Networks and Heterogeneous Media, 2008, 3(1): 43-83. doi: 10.3934/nhm.2008.3.43

    Related Papers:

  • In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth of the tumor mass. By weakening the role of the extracellular matrix, which is regarded as a rigid non-remodeling scaffold, a system of two partial differential equations is derived, describing the evolution of the cell volume ratio coupled to the dynamics of the nutrient, whose higher and lower concentration levels determine proliferation or death of tumor cells, respectively. Numerical simulations of a reference two-dimensional problem are shown and commented, and a qualitative mathematical analysis of some of its key issues is proposed.


  • This article has been cited by:

    1. Hermann B. Frieboes, Fang Jin, Yao-Li Chuang, Steven M. Wise, John S. Lowengrub, Vittorio Cristini, Three-dimensional multispecies nonlinear tumor growth—II: Tumor invasion and angiogenesis, 2010, 264, 00225193, 1254, 10.1016/j.jtbi.2010.02.036
    2. Ying Chen, Steven M. Wise, Vivek B. Shenoy, John S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, 2014, 30, 20407939, 726, 10.1002/cnm.2624
    3. Dylan A. Goodin, Hermann B. Frieboes, Simulation of 3D centimeter-scale continuum tumor growth at sub-millimeter resolution via distributed computing, 2021, 134, 00104825, 104507, 10.1016/j.compbiomed.2021.104507
    4. J S Lowengrub, H B Frieboes, F Jin, Y-L Chuang, X Li, P Macklin, S M Wise, V Cristini, Nonlinear modelling of cancer: bridging the gap between cells and tumours, 2010, 23, 0951-7715, R1, 10.1088/0951-7715/23/1/R01
    5. Ying Chen, John S. Lowengrub, Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach, 2014, 361, 00225193, 14, 10.1016/j.jtbi.2014.06.024
    6. Luigi Preziosi, Andrea Tosin, Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications, 2009, 58, 0303-6812, 625, 10.1007/s00285-008-0218-7
    7. Riccardo Sacco, Paola Causin, Chiara Lelli, Manuela T. Raimondi, A poroelastic mixture model of mechanobiological processes in biomass growth: theory and application to tissue engineering, 2017, 52, 0025-6455, 3273, 10.1007/s11012-017-0638-9
    8. S.N. Antontsev, А.А. Papin, M.A. Tokareva, E.I. Leonova, E.A. Gridushko, Modeling of Tumor Occurrence and Growth – II, 2021, 1561-9451, 72, 10.14258/izvasu(2021)1-12
    9. Dylan A. Goodin, Hermann B. Frieboes, Evaluation of innate and adaptive immune system interactions in the tumor microenvironment via a 3D continuum model, 2023, 559, 00225193, 111383, 10.1016/j.jtbi.2022.111383
    10. S.M. Wise, J.S. Lowengrub, V. Cristini, An adaptive multigrid algorithm for simulating solid tumor growth using mixture models, 2011, 53, 08957177, 1, 10.1016/j.mcm.2010.07.007
    11. Andrea Tosin, Initial/boundary-value problems of tumor growth within a host tissue, 2013, 66, 0303-6812, 163, 10.1007/s00285-012-0505-1
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  • © 2008 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)
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