Mathematical Biosciences and Engineering, 2005, 2(3): 643-655. doi: 10.3934/mbe.2005.2.643.

92C10, 92C37, 92C50, 76D05, 76M20.

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A Single-Cell Approach in Modeling the Dynamics of Tumor Microregions

1. Mathematical Biosciences Institute, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210

Interactions between tumor cells and their environment lead to the formation of microregions containing nonhomogeneous subpopulations of cells and steep gradients in oxygen, glucose, and other metabolites. To address the formation of tumor microregions on the level of single cells, I propose a new two-dimensional time-dependent mathematical model taking explicitly into account the individually regulated biomechanical processes of tumor cells and the effect of oxygen consumption on their metabolism. Numerical simulations of the self-organized formation of tumor microregions are presented and the dynamics of such a process is discussed.
  Article Metrics

Keywords avascular tumor growth; mathe- matical modeling; tumor microregions development; immersed boundary method.

Citation: Katarzyna A. Rejniak. A Single-Cell Approach in Modeling the Dynamics of Tumor Microregions. Mathematical Biosciences and Engineering, 2005, 2(3): 643-655. doi: 10.3934/mbe.2005.2.643


This article has been cited by

  • 1. P. Van Liedekerke, M. M. Palm, N. Jagiella, D. Drasdo, Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results, Computational Particle Mechanics, 2015, 2, 4, 401, 10.1007/s40571-015-0082-3
  • 2. Fergus R. Cooper, Ruth E. Baker, Alexander G. Fletcher, Numerical Analysis of the Immersed Boundary Method for Cell-Based Simulation, SIAM Journal on Scientific Computing, 2017, 39, 5, B943, 10.1137/16M1092246
  • 3. YANGJIN KIM, MAGDALENA A. STOLARSKA, HANS G. OTHMER, A HYBRID MODEL FOR TUMOR SPHEROID GROWTH IN VITRO I: THEORETICAL DEVELOPMENT AND EARLY RESULTS, Mathematical Models and Methods in Applied Sciences, 2007, 17, supp01, 1773, 10.1142/S0218202507002479
  • 4. Nikodem J. Popławski, Ubirajara Agero, J. Scott Gens, Maciej Swat, James A. Glazier, Alexander R. A. Anderson, Front Instabilities and Invasiveness of Simulated Avascular Tumors, Bulletin of Mathematical Biology, 2009, 71, 5, 1189, 10.1007/s11538-009-9399-5
  • 5. Daniel K. Wells, Yishan Chuang, Louis M. Knapp, Dirk Brockmann, William L. Kath, Joshua N. Leonard, Martin Meier-Schellersheim, Spatial and Functional Heterogeneities Shape Collective Behavior of Tumor-Immune Networks, PLOS Computational Biology, 2015, 11, 4, e1004181, 10.1371/journal.pcbi.1004181
  • 6. Yoonseok Kam, Katarzyna A. Rejniak, Alexander R.A. Anderson, Cellular modeling of cancer invasion: Integration of in silico and in vitro approaches, Journal of Cellular Physiology, 2012, 227, 2, 431, 10.1002/jcp.22766
  • 7. Katarzyna A. Rejniak, An immersed boundary framework for modelling the growth of individual cells: An application to the early tumour development, Journal of Theoretical Biology, 2007, 247, 1, 186, 10.1016/j.jtbi.2007.02.019
  • 8. S.M. Wise, J.S. Lowengrub, V. Cristini, An adaptive multigrid algorithm for simulating solid tumor growth using mixture models, Mathematical and Computer Modelling, 2011, 53, 1-2, 1, 10.1016/j.mcm.2010.07.007
  • 9. Katarzyna A. Rejniak, Alexander R. A. Anderson, A Computational Study of the Development of Epithelial Acini: II. Necessary Conditions for Structure and Lumen Stability, Bulletin of Mathematical Biology, 2008, 70, 5, 1450, 10.1007/s11538-008-9308-3
  • 10. Nara Yoon, Robert Vander Velde, Andriy Marusyk, Jacob G. Scott, Optimal Therapy Scheduling Based on a Pair of Collaterally Sensitive Drugs, Bulletin of Mathematical Biology, 2018, 80, 7, 1776, 10.1007/s11538-018-0434-2
  • 11. Katarzyna A. Rejniak, Alexander R. A. Anderson, A Computational Study of the Development of Epithelial Acini: I. Sufficient Conditions for the Formation of a Hollow Structure, Bulletin of Mathematical Biology, 2008, 70, 3, 677, 10.1007/s11538-007-9274-1
  • 12. P. Gerlee, A.R.A. Anderson, Evolution of cell motility in an individual-based model of tumour growth, Journal of Theoretical Biology, 2009, 259, 1, 67, 10.1016/j.jtbi.2009.03.005
  • 13. G. U. Unnikrishnan, V. U. Unnikrishnan, J. N. Reddy, C. T. Lim, Review on the Constitutive Models of Tumor Tissue for Computational Analysis, Applied Mechanics Reviews, 2010, 63, 4, 040801, 10.1115/1.4002427
  • 14. Nikodem J. Poplawski, Abbas Shirinifard, Ubirajara Agero, J. Scott Gens, Maciej Swat, James A. Glazier, Gustavo Stolovitzky, Front Instabilities and Invasiveness of Simulated 3D Avascular Tumors, PLoS ONE, 2010, 5, 5, e10641, 10.1371/journal.pone.0010641
  • 15. K. Saetzler, C. Sonnenschein, A.M. Soto, Systems biology beyond networks: Generating order from disorder through self-organization, Seminars in Cancer Biology, 2011, 21, 3, 165, 10.1016/j.semcancer.2011.04.004
  • 16. EUNOK JUNG, DO WAN KIM, JONGGUL LEE, WANHO LEE, MULTIDIMENSIONAL OPEN SYSTEM FOR VALVELESS PUMPING, Bulletin of the Korean Mathematical Society, 2015, 52, 6, 1973, 10.4134/BKMS.2015.52.6.1973
  • 17. Yibao Li, Junseok Kim, Three-dimensional simulations of the cell growth and cytokinesis using the immersed boundary method, Mathematical Biosciences, 2016, 271, 118, 10.1016/j.mbs.2015.11.005
  • 18. Maymona Al-Husari, Craig Murdoch, Steven D. Webb, A cellular automaton model examining the effects of oxygen, hydrogen ions and lactate on early tumour growth, Journal of Mathematical Biology, 2014, 69, 4, 839, 10.1007/s00285-013-0719-x
  • 19. Alexander G. Fletcher, Fergus Cooper, Ruth E. Baker, Mechanocellular models of epithelial morphogenesis, Philosophical Transactions of the Royal Society B: Biological Sciences, 2017, 372, 1720, 20150519, 10.1098/rstb.2015.0519
  • 20. Chun-Chao Wang, Leen Jamal, Kevin A. Janes, Normal morphogenesis of epithelial tissues and progression of epithelial tumors, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 2012, 4, 1, 51, 10.1002/wsbm.159
  • 21. Andreas Deutsch, Sabine Dormann, , Cellular Automaton Modeling of Biological Pattern Formation, 2017, Chapter 14, 347, 10.1007/978-1-4899-7980-3_14
  • 22. Jonathan F. Li, John Lowengrub, The effects of cell compressibility, motility and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model, Journal of Theoretical Biology, 2014, 343, 79, 10.1016/j.jtbi.2013.10.008
  • 23. Fabiano L. Ribeiro, Kayo N. Ribeiro, A one dimensional model of population growth, Physica A: Statistical Mechanics and its Applications, 2015, 434, 201, 10.1016/j.physa.2015.03.021
  • 24. Yibao Li, Ana Yun, Junseok Kim, An immersed boundary method for simulating a single axisymmetric cell growth and division, Journal of Mathematical Biology, 2012, 65, 4, 653, 10.1007/s00285-011-0476-7
  • 25. Katarzyna A. Rejniak, Alexander R. A. Anderson, Hybrid models of tumor growth, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 2011, 3, 1, 115, 10.1002/wsbm.102
  • 26. Katarzyna A. Rejniak, Shizhen E. Wang, Nicole S. Bryce, Hang Chang, Bahram Parvin, Jerome Jourquin, Lourdes Estrada, Joe W. Gray, Carlos L. Arteaga, Alissa M. Weaver, Vito Quaranta, Alexander R. A. Anderson, Andrew D. McCulloch, Linking Changes in Epithelial Morphogenesis to Cancer Mutations Using Computational Modeling, PLoS Computational Biology, 2010, 6, 8, e1000900, 10.1371/journal.pcbi.1000900
  • 27. Sunmi Lee, Eunok Jung, A two-chamber model of valveless pumping using the immersed boundary method, Applied Mathematics and Computation, 2008, 206, 2, 876, 10.1016/j.amc.2008.09.047
  • 28. P. Van Liedekerke, A. Buttenschön, D. Drasdo, , Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, 2018, 245, 10.1016/B978-0-12-811718-7.00014-9
  • 29. 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, Nonlinearity, 2010, 23, 1, R1, 10.1088/0951-7715/23/1/R01
  • 30. Jia Zhao, Qi Wang, A 3D Multi-Phase Hydrodynamic Model for Cytokinesis of Eukaryotic Cells, Communications in Computational Physics, 2016, 19, 03, 663, 10.4208/cicp.181014.140715a
  • 31. Jia Zhao, Qi Wang, Modeling cytokinesis of eukaryotic cells driven by the actomyosin contractile ring, International Journal for Numerical Methods in Biomedical Engineering, 2016, 32, 12, e02774, 10.1002/cnm.2774
  • 32. Alexander R. A. Anderson, Katarzyna A. Rejniak, Philip Gerlee, Vito Quaranta, Microenvironment driven invasion: a multiscale multimodel investigation, Journal of Mathematical Biology, 2009, 58, 4-5, 579, 10.1007/s00285-008-0210-2
  • 33. Timothy J. Newman, , Single-Cell-Based Models in Biology and Medicine, 2007, Chapter 10, 221, 10.1007/978-3-7643-8123-3_10
  • 34. Katarzyna A. Rejniak, , Single-Cell-Based Models in Biology and Medicine, 2007, Chapter 13, 301, 10.1007/978-3-7643-8123-3_13
  • 35. Alexander R. A. Anderson, , Selected Topics in Cancer Modeling, 2008, Chapter 11, 1, 10.1007/978-0-8176-4713-1_11
  • 36. Katarzyna A. Rejniak, , Discrete and Topological Models in Molecular Biology, 2014, Chapter 23, 507, 10.1007/978-3-642-40193-0_23
  • 37. Krzysztof Psiuk-Maksymowicz, Damian Borys, Sebastian Student, Andrzej Świerniak, , Information Technologies in Biomedicine, Volume 3, 2014, Chapter 23, 261, 10.1007/978-3-319-06593-9_23

Reader Comments

your name: *   your email: *  

Copyright Info: 2005, Katarzyna A. Rejniak, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved