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THE METHOD OF LINES FOR THE NUMERICAL SOLUTION OF A MATHEMATICAL MODEL IN THE INITIATION OF ANGIOGENESIS

Year 2013, Volume 3, Issue 2, 182 - 197, 01.12.2013

Abstract

In this paper we present the method of lines to obtain the numerical solution of a mathematical model for the roles of endothelial, pericyte and macrophage cells in the initiation of tumor angiogenesis. This method is an approach to the numerical solution of partial differential equations that involve a time variable t and a space variable x. We provide computer programs written in Matlab. Also, the stability analysis of the solutions is given, and the figures for endothelial, pericyte and macrophage cell movements in the capillary are presented for large times

References

  • Ahmad, I. and Berzins, M., (2001), MOL solvers for hyperbolic PDEs with source terms, Math. Comput. Simulat., 56 (2), 115-125.
  • Erdem, A. and Pamuk, S., (2007), The method of lines for the numerical solution of a mathematical model for capillary formation: The role of tumor angiogenic factor in the extra-cellular matrix, Appl. Math. Comput., 186, 891-897.
  • Korn, G.A., (1999), Interactive solution of partial differential equations by the Method-of-lines, Math. Comput. Simulat., 49 (1-2), 129-138.
  • Levine, H.A., Sleeman, B.D. and Nilsen-Hamilton, M., (2000), A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I.The role of protease inhibitors in preventing angiogenesis, Math. Biosc., 168, 77-115.
  • Mikhail, M.N., (1987), On the validity and stability of the method of lines for the solution of partial differential equations, Appl. Math. Comput., 22 (2-3), 89-98.
  • Pamuk, S. and Erdem, A., (2007), The method of lines for the numerical solution of a mathematical model for capillary formation: The role of endothelial cells in the capillary, Appl. Math. Comput., 186, 831-835.
  • Pamuk, S., (2003), Qualitative analysis of a mathematical model for capillary formation in tumor angiogenesis, Math. Models and Methods Apll. Sci., 13 (1), 19-33.
  • Schor, A.M., CanŞeld, A.E., Sutton,A.B., Allen,T.D., Sloan,P., Schor, S.L., Steiner,R., Weisz, P.B. and Langer,R., (1992), The behavior of pericytes in vitro: relevance to angiogenesis and differentiation, Angiogenesis: Key Principles-Science-Technology-Medicine, Birkhauser, Basel.
  • Shakeri, F. and Dehghan, M., (2008), The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition, Comput. and Math. with Appl., 56, 2175-2188. [10] Sharaf, A.A. and Bakodah, H.O., (2005), A good spatial discretisation in the method of lines, Appl. Math. Comput., 171 (2), 1253-1263.
  • Younes,A., Konz,M., Fahs, M., Zidane, A. and Huggenberger, P., (2011), Modelling variable den- sity flow problems in heterogeneous porous media using the method of lines and advanced spatial discretization methods, Math. Comput. Simulat., 81, 2346-2355.
  • Teschl, G., (1991), Ordinary Differential Equations and Dynamical Systems, American Mathematical Society.

Year 2013, Volume 3, Issue 2, 182 - 197, 01.12.2013

Abstract

References

  • Ahmad, I. and Berzins, M., (2001), MOL solvers for hyperbolic PDEs with source terms, Math. Comput. Simulat., 56 (2), 115-125.
  • Erdem, A. and Pamuk, S., (2007), The method of lines for the numerical solution of a mathematical model for capillary formation: The role of tumor angiogenic factor in the extra-cellular matrix, Appl. Math. Comput., 186, 891-897.
  • Korn, G.A., (1999), Interactive solution of partial differential equations by the Method-of-lines, Math. Comput. Simulat., 49 (1-2), 129-138.
  • Levine, H.A., Sleeman, B.D. and Nilsen-Hamilton, M., (2000), A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I.The role of protease inhibitors in preventing angiogenesis, Math. Biosc., 168, 77-115.
  • Mikhail, M.N., (1987), On the validity and stability of the method of lines for the solution of partial differential equations, Appl. Math. Comput., 22 (2-3), 89-98.
  • Pamuk, S. and Erdem, A., (2007), The method of lines for the numerical solution of a mathematical model for capillary formation: The role of endothelial cells in the capillary, Appl. Math. Comput., 186, 831-835.
  • Pamuk, S., (2003), Qualitative analysis of a mathematical model for capillary formation in tumor angiogenesis, Math. Models and Methods Apll. Sci., 13 (1), 19-33.
  • Schor, A.M., CanŞeld, A.E., Sutton,A.B., Allen,T.D., Sloan,P., Schor, S.L., Steiner,R., Weisz, P.B. and Langer,R., (1992), The behavior of pericytes in vitro: relevance to angiogenesis and differentiation, Angiogenesis: Key Principles-Science-Technology-Medicine, Birkhauser, Basel.
  • Shakeri, F. and Dehghan, M., (2008), The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition, Comput. and Math. with Appl., 56, 2175-2188. [10] Sharaf, A.A. and Bakodah, H.O., (2005), A good spatial discretisation in the method of lines, Appl. Math. Comput., 171 (2), 1253-1263.
  • Younes,A., Konz,M., Fahs, M., Zidane, A. and Huggenberger, P., (2011), Modelling variable den- sity flow problems in heterogeneous porous media using the method of lines and advanced spatial discretization methods, Math. Comput. Simulat., 81, 2346-2355.
  • Teschl, G., (1991), Ordinary Differential Equations and Dynamical Systems, American Mathematical Society.

Details

Primary Language Turkish
Journal Section Research Article
Authors

S. PAMUK This is me
Department of Mathematics, University of Kocaeli, Umuttepe Campus, 41380, Kocaeli - TURKEY


I. ATAC This is me
Department of Mathematics, University of Kocaeli, Umuttepe Campus, 41380, Kocaeli - TURKEY

Publication Date December 1, 2013
Published in Issue Year 2013, Volume 3, Issue 2

Cite

Bibtex @ { twmsjaem761409, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2013}, volume = {3}, number = {2}, pages = {182 - 197}, title = {THE METHOD OF LINES FOR THE NUMERICAL SOLUTION OF A MATHEMATICAL MODEL IN THE INITIATION OF ANGIOGENESIS}, key = {cite}, author = {Pamuk, S. and Atac, I.} }