Numerical Solutions for a Model of Tissue Invasion and Migration of Tumour Cells

The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points. Our numerical experiments illustrate the metastatic ability of tumour cells.

[1]  M. Chaplain,et al.  Mathematical Modelling of Tissue Invasion , 2003 .

[2]  M Tubiana,et al.  Measuring progress against cancer in Europe: has the 15% decline targeted for 2000 come about? , 2003, Annals of oncology : official journal of the European Society for Medical Oncology.

[3]  E. T. Gawlinski,et al.  A reaction-diffusion model of cancer invasion. , 1996, Cancer research.

[4]  Robert N. Hughes,et al.  Cancer: Principles and Practice of Oncology , 2005 .

[5]  P. Steeg,et al.  Mechanisms of tumor invasion and metastasis , 2004, World Journal of Urology.

[6]  Nicola Bellomo,et al.  Modelling Complex systems in Molecular Biology and Tumor Dynamics and Control , 2005 .

[7]  U. Veronesi,et al.  Oxford textbook of oncology , 1996 .

[8]  Mark A. J. Chaplain,et al.  Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity , 2006, Networks Heterog. Media.

[9]  P. Maini,et al.  MODELLING THE RESPONSE OF VASCULAR TUMOURS TO CHEMOTHERAPY: A MULTISCALE APPROACH , 2006 .

[10]  T. Vincent,et al.  EVOLUTIONARY DYNAMICS IN CARCINOGENESIS , 2005 .

[11]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[12]  C. Bangham,et al.  Review. An introduction to lymphocyte and viral dynamics: the power and limitations of mathematical analysis , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[13]  A. J. Perumpanani,et al.  Traveling Shock Waves Arising in a Model of Malignant Invasion , 1999, SIAM J. Appl. Math..

[14]  L. Liotta,et al.  Tumor invasion and the extracellular matrix. , 1983, Laboratory investigation; a journal of technical methods and pathology.

[15]  B Møller,et al.  Cancer mortality trends in the EU and acceding countries up to 2015. , 2003, Annals of oncology : official journal of the European Society for Medical Oncology.

[16]  H. Othmer,et al.  Mathematical modeling of tumor-induced angiogenesis , 2004, Journal of mathematical biology.

[17]  N. Bellomo,et al.  On the onset of non-linearity for diffusion models of binary mixtures of biological materials by asymptotic analysis , 2006 .

[18]  H. Byrne,et al.  Extracellular matrix concentration exerts selection pressure on invasive cells. , 1999, European journal of cancer.

[19]  R. A. ANDERSONa,et al.  Mathematical Modelling of Tumour Invasion and Metastasis , 2022 .

[20]  Helen M. Byrne,et al.  A Mathematical Model of Trophoblast Invasion , 1999 .

[21]  H. Bartelink,et al.  European Code Against Cancer and scientific justification: third version (2003). , 2003, Annals of oncology : official journal of the European Society for Medical Oncology.

[22]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

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

[24]  A. d’Onofrio.  TUMOR-IMMUNE SYSTEM INTERACTION: MODELING THE TUMOR-STIMULATED PROLIFERATION OF EFFECTORS AND IMMUNOTHERAPY , 2006 .

[25]  Ramesh Srinivasan,et al.  Path-integral evolution of chaos embedded in noise: Duffing neocortical analog , 1996 .

[26]  T. Wheldon.  Mathematical models in cancer research , 1988 .

[27]  Francesco Pappalardo,et al.  MODELING TUMOR IMMUNOLOGY , 2006 .

[28]  A. Anderson,et al.  A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. , 2005, Mathematical medicine and biology : a journal of the IMA.

[29]  Cheryl Jorcyk,et al.  Alterations in gene expression profiles during prostate cancer progression: functional correlations to tumorigenicity and down-regulation of selenoprotein-P in mouse and human tumors. , 2002, Cancer research.

[30]  Mark A. J. Chaplain,et al.  A mathematical model of vascular tumour growth and invasion , 1996 .

[31]  M. Lachowicz.  MICRO AND MESO SCALES OF DESCRIPTION CORRESPONDING TO A MODEL OF TISSUE INVASION BY SOLID TUMOURS , 2005 .

[32]  Helen M. Byrne,et al.  An analysis of a mathematical model of trophoblast invasion , 2001, Appl. Math. Lett..

[33]  Alberto d'Onofrio,et al.  Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy , 2008, Math. Comput. Model..

[34]  T. P. Pretlow,et al.  Differential experimental micrometastasis to lung, liver, and bone with lacZ-tagged CWR22R prostate carcinoma cells , 2004, Clinical & Experimental Metastasis.

[35]  J. Sherratt,et al.  Extracellular matrix-mediated chemotaxis can impede cell migration , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[36]  M. Kolev,et al.  Numerical Versus Experimental Data for Prostate Tumour Growth , 2011 .

[37]  R. Gatenby,et al.  Models of tumor-host interaction as competing populations: implications for tumor biology and treatment. , 1995, Journal of theoretical biology.

[38]  L. Orci,et al.  The receptor for urokinase type plasminogen activator polarizes expression of the protease to the leading edge of migrating monocytes and promotes degradation of enzyme inhibitor complexes , 1990, The Journal of cell biology.

[39]  L. Preziosi,et al.  Modelling and mathematical problems related to tumor evolution and its interaction with the immune system , 2000 .

[40]  E. Klein,et al.  Biology of metastasis and its clinical implications: renal-cell cancer , 2004, World Journal of Urology.

[41]  L. Liotta,et al.  Tumor cell interactions with the extracellular matrix during invasion and metastasis. , 1993, Annual review of cell biology.

[42]  J. Sherratt,et al.  A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion , 1999 .

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