A Multiscale Approach to the Migration of Cancer Stem Cells: Mathematical Modelling and Simulations

We propose a multiscale model for the invasion of the extracellular matrix by two types of cancer cells, the differentiated cancer cells and the cancer stem cells. We investigate the epithelial mesenchymal-like transition between them being driven primarily by the epidermal growth factors. We moreover take into account the transdifferentiation program of the cancer stem cells towards the cancer-associated fibroblast cells as well as the fibroblast-driven remodelling of the extracellular matrix. The proposed haptotaxis model combines the macroscopic phenomenon of the invasion of the extracellular matrix by both types of cancer cells with the microscopic dynamics of the epidermal growth factors. We analyse our model in a component-wise manner and compare our findings with the literature. We investigate pathological situations regarding the epidermal growth factors and accordingly propose “mathematical-treatment” scenarios to control the aggressiveness of the tumour.

[1]  W. Lam,et al.  Acquired resistance to EGFR inhibitors is associated with a manifestation of stem cell-like properties in cancer cells. , 2013, Cancer research.

[2]  Nikolaos Sfakianakis,et al.  A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix , 2015, 1505.04268.

[3]  Stuart K. Williams,et al.  Migration of individual microvessel endothelial cells: stochastic model and parameter measurement. , 1991, Journal of cell science.

[4]  D A Lauffenburger,et al.  Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. , 1991, Journal of theoretical biology.

[5]  Franziska Michor,et al.  Mathematical models of cancer stem cells. , 2008, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[6]  G. Barlow,et al.  FURTHER STUDIES ON THE PURIFICATION AND CHARACTERIZATION OF HUMAN PLASMINOGEN AND PLASMIN. , 1965, The Journal of biological chemistry.

[7]  J. C. FISHER,et al.  Multiple-Mutation Theory of Carcinogenesis , 1958, Nature.

[8]  Mark A. J. Chaplain,et al.  Robust numerical methods for taxis-diffusion-reaction systems: Applications to biomedical problems , 2006, Math. Comput. Model..

[9]  Peter Klein,et al.  A structure-based model for ligand binding and dimerization of EGF receptors. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Thomas R. Cox,et al.  Remodeling and homeostasis of the extracellular matrix: implications for fibrotic diseases and cancer , 2011, Disease Models & Mechanisms.

[11]  Alexander Kurganov,et al.  Numerical study of two-species chemotaxis models , 2013 .

[12]  Mark A. J. Chaplain,et al.  Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation , 2011, Journal of mathematical biology.

[13]  R. Weinberg,et al.  Cancer stem cells: mirage or reality? , 2009, Nature Medicine.

[14]  Dennis Bray,et al.  Cell Movements: From Molecules to Motility , 1992 .

[15]  Thomas Kirchner,et al.  Migrating cancer stem cells — an integrated concept of malignant tumour progression , 2005, Nature Reviews Cancer.

[16]  Valerie M. Weaver,et al.  Three-dimensional context regulation of metastasis , 2008, Clinical & Experimental Metastasis.

[17]  M. Chaplain,et al.  Mathematical modelling of cancer cell invasion of tissue , 2005, Math. Comput. Model..

[18]  C. Nordling.  A New Theory on the Cancer-inducing Mechanism , 1953, British Journal of Cancer.

[19]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[20]  P. Armitage,et al.  The age distribution of cancer and a multi-stage theory of carcinogenesis , 1954, British Journal of Cancer.

[21]  Peter Klein,et al.  On the nature of low- and high-affinity EGF receptors on living cells. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[22]  M. Chaplain,et al.  Mathematical modelling of tumour invasion and metastasis , 2000 .

[23]  D. Radisky.  Epithelial-mesenchymal transition , 2005, Journal of Cell Science.

[24]  M. Noble,et al.  Cancer stem cells. , 2006, The New England journal of medicine.

[25]  Kshitiz Gupta,et al.  Mechanosensitivity of fibroblast cell shape and movement to anisotropic substratum topography gradients. , 2009, Biomaterials.

[26]  Y. Yamashita,et al.  The Role of Epidermal Growth Factor Receptor in Cancer Metastasis and Microenvironment , 2013, BioMed research international.

[27]  K. Painter,et al.  Mathematical modelling of glioma growth: the use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion. , 2013, Journal of theoretical biology.

[28]  Yuri Kogan,et al.  Strategies for cancer stem cell elimination: insights from mathematical modeling. , 2012, Journal of theoretical biology.

[29]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[30]  Luigi Preziosi,et al.  Cancer Modelling and Simulation , 2003 .

[31]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[32]  Philip K Maini,et al.  On the proportion of cancer stem cells in a tumour. , 2010, Journal of theoretical biology.

[33]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection , 1977 .

[34]  Wenjun Guo,et al.  The Epithelial-Mesenchymal Transition Generates Cells with Properties of Stem Cells , 2008, Cell.

[35]  Avner Friedman,et al.  Interaction of Tumor with Its Micro-environment: A Mathematical Model , 2010, Bulletin of mathematical biology.

[36]  V. Seltzer,et al.  EGF mediates calcium-activated chloride channel activation in the human bronchial epithelial cell line 16HBE14o-: involvement of tyrosine kinase p60c-src. , 2008, American journal of physiology. Lung cellular and molecular physiology.

[37]  A. Marciniak-Czochra,et al.  Mathematical Modeling of Leukemogenesis and Cancer Stem Cell Dynamics , 2012 .

[38]  Xin-hua Liang,et al.  A new perspective of vasculogenic mimicry: EMT and cancer stem cells (Review) , 2013, Oncology letters.

[39]  John Mendelsohn,et al.  The EGF receptor family as targets for cancer therapy , 2000, Oncogene.

[40]  J. Brandts.  [Review of: W. Hundsdorfer, J.G. Verwer (2003) Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations] , 2006 .

[41]  David Isaacson,et al.  A Mathematical Model for the Effects of HER2 Overexpression on Cell Proliferation in Breast Cancer , 2008, Bulletin of mathematical biology.

[42]  G. Russo,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2005 .

[43]  J. Sherratt,et al.  Biological inferences from a mathematical model for malignant invasion. , 1996, Invasion & metastasis.

[44]  Fazlul H. Sarkar,et al.  Cancer Stem Cells and Epithelial-to-Mesenchymal Transition (EMT)-Phenotypic Cells: Are They Cousins or Twins? , 2011, Cancers.

[45]  Zang Ai-hua,et al.  Stem Cells,Cancer and Cancer Stem Cells , 2005 .

[46]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[47]  M. Chaplain,et al.  Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. , 2008 .

[48]  D A Lauffenburger,et al.  Transient behavior of a chemotaxis system modelling certain types of tissue inflammation , 1987, Journal of mathematical biology.

[49]  T. Tan,et al.  EMT-Induced Stemness and Tumorigenicity Are Fueled by the EGFR/Ras Pathway , 2013, PloS one.

[50]  Nikolaos Sfakianakis,et al.  A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model , 2016, Appl. Math. Comput..

[51]  R. Ganguly,et al.  Mathematical model for the cancer stem cell hypothesis , 2006, Cell proliferation.

[52]  J. Mendelsohn,et al.  Relation of epidermal growth factor receptor concentration to growth of human epidermoid carcinoma A431 cells. , 1984, The Journal of biological chemistry.

[53]  J. Thiery.  Epithelial–mesenchymal transitions in tumour progression , 2002, Nature Reviews Cancer.

[54]  Nicola Bellomo,et al.  On the foundations of cancer modelling: Selected topics, speculations, and perspectives , 2008 .

[55]  Christina Surulescu,et al.  Global existence for a go-or-grow multiscale model for tumor invasion with therapy , 2016 .

[56]  Ling Xia,et al.  Cancer stem cell, niche and EGFR decide tumor development and treatment response: A bio-computational simulation study. , 2011, Journal of theoretical biology.

[57]  Roger R. Markwald,et al.  Computational modeling of epithelial-mesenchymal transformations , 2010, Biosyst..

[58]  Y. Imai,et al.  Epidermal growth factor receptors and effect of epidermal growth factor on growth of human breast cancer cells in long-term tissue culture. , 1982, Cancer research.

[59]  Stephen T. C. Wong,et al.  Microenvironmental regulation of epithelial-mesenchymal transitions in cancer. , 2012, Cancer research.

[60]  Alf Gerisch,et al.  Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. , 2014, Journal of theoretical biology.

[61]  Samy Lamouille,et al.  TGF-&bgr; signaling and epithelial–mesenchymal transition in cancer progression , 2013, Current opinion in oncology.

[62]  S. McDougall,et al.  Mathematical modelling of flow through vascular networks: Implications for tumour-induced angiogenesis and chemotherapy strategies , 2002, Bulletin of mathematical biology.

[63]  S. Rafii,et al.  VEGFR1-positive haematopoietic bone marrow progenitors initiate the pre-metastatic niche , 2005, Nature.