Analysis on a Mathematical Model for Tumor Induced Angiogenesis
Affiliation(s)
Department of Computational Science & Mathematics, Middle Tennessee State University, Murfreesboro, USA.
Department of Computational Science & Mathematics, Middle Tennessee State University, Murfreesboro, USA.
ABSTRACT
Tumor-induced angiogenesis is the process by which unmetastasized tumors recruit red blood vessels by way of chemical stimuli to grow towards the tumor for vascularization and metastasis. We model the process of tumor-induced angiogenesis at the tissue level using ordinary and partial differential equations (ODEs and PDEs) that have a source term. The source term is associated with a signal for growth factors from the tumor. We assume that the source term depends on time, and a parameter (time parameter). We use an explicit stabilized Runge-Kutta method to solve the partial differential equation. By introducing a source term into the PDE model, we extend the PDE model used by H. A. Harrington et al. Our results suggest that the time parameter could play some role in understanding angiogenesis.
Cite this paper
Ewool, R. and Sinkala, Z. (2014) Analysis on a Mathematical Model for Tumor Induced Angiogenesis. Journal of Applied Mathematics and Physics, 2, 698-707. doi: 10.4236/jamp.2014.27077.
Ewool, R. and Sinkala, Z. (2014) Analysis on a Mathematical Model for Tumor Induced Angiogenesis. Journal of Applied Mathematics and Physics, 2, 698-707. doi: 10.4236/jamp.2014.27077.
References
[1] Benjamin Ribba et al. (2006) A Multiscale Mathematical Model of Avascular Tumor Growth to Investigate the Therapeutic Benefit of Anti-Invasive Agents. Elsevier Science, Preprint. [2] Pamuk, S. (2007) A Mathematical Model for Tumor Angiogenesis. University of Idaho, Moscow, Idaho. [3] Anderson, A. and Chaplain, M. (1998) Continuous and Discrete Mathematical Models of Tumor-Induced Angiogenesis. Bulletin of Mathematical Biology, 60, 857-899. http://dx.doi.org/10.1006/bulm.1998.0042 [4] Levine, H., Tucker, A. and Nilsen-Hamilton, M. (2002) A Mathematical Model for the Role of Cell Signal Transduction in the Initiation and Inhibition of Angiogenesis. Growth Factors, 20, 155-175.
http://dx.doi.org/10.1080/0897719031000084355 [5] Good, D., Polverini, P., Rastinejad, F., Le Beau, M., Lemons, R., Frazier, W. and Bouck, N. (1990) A Tumor Suppressor-Dependent Inhibitor of Angiogenesis Is Immunologically and Functionally Indistinguishable from a Fragment of Thrombospondin. Proceedings of the National Academy of Sciences, 87, 6624-6628.
http://dx.doi.org/10.1073/pnas.87.17.6624 [6] Bickel, S.T., Juliano, J.D. and Nagy, J.D. (2014) Evolution of Proliferation and the Angiogenic Switch in Tumors with High Clonal Diversity. PLoS ONE, 9, Article ID: e91992. [7] Folkman, J. (1971) Fundamental Concept of the Angiogenic Process. The New England Journal of Medicine, 285, 1182-1186. [8] Harrington, H.A., Maier, M., Naidoo, L., Whitaker, N. and Kevrekidis, P.G. (2006) A Hybrid Model for Tumor-Induced Angiogenesis in the Cornea in the Presence of Inhibitors. Mathematical and Computer Modelling, 46, 513-524.
http://dx.doi.org/10.1016/j.mcm.2006.11.034 [9] Tong, S. and Yuan, F. (2001) Numerical Simualtion of Angiogeneis in the Cornea. Microvascular Research, 61, 14-27. [10] Abdulle, A. (2002) Fourth Order Chebyshev Methods with Recurrence Relation. SIAM Journal on Scientific Computing, 23, 2041-2054. http://dx.doi.org/10.1137/S1064827500379549 [11] Van der Houwen, S.B, (1980) On the Internal Stage Runge-Kutta Methods for Large m-Values. Zeitschrift für Angewandte Mathematik und Mechanik, 60, 479-485. [12] Verwer, J.G. (1996) Explicit Runge-Kutta Methods for Parabolic Differential Equations. Applied Numerical Mathematics, 22, 359-379. http://dx.doi.org/10.1016/S0168-9274(96)00022-0 [13] Sommeijer, B., Shampine, L. and Verwer, J. (1998) RKC: An Explicit Solver for Parabolic PDEs. Journal of Computational and Applied Mathematics, 88, 326. [14] Riha, W. (1972) Optimal Stability Polynominals. Computing, 9, 37-43.
http://dx.doi.org/10.1007/BF02236374 [15] Abdulle, A. (2011) Explicit Stabilized Runge-Kutta Methods. MATHICSE Technical Report, Nr. 27.2011. [16] LeVeque, R.J. (2007) Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 175-179. http://dx.doi.org/10.1007/BF02236374
[1] Benjamin Ribba et al. (2006) A Multiscale Mathematical Model of Avascular Tumor Growth to Investigate the Therapeutic Benefit of Anti-Invasive Agents. Elsevier Science, Preprint. [2] Pamuk, S. (2007) A Mathematical Model for Tumor Angiogenesis. University of Idaho, Moscow, Idaho. [3] Anderson, A. and Chaplain, M. (1998) Continuous and Discrete Mathematical Models of Tumor-Induced Angiogenesis. Bulletin of Mathematical Biology, 60, 857-899. http://dx.doi.org/10.1006/bulm.1998.0042 [4] Levine, H., Tucker, A. and Nilsen-Hamilton, M. (2002) A Mathematical Model for the Role of Cell Signal Transduction in the Initiation and Inhibition of Angiogenesis. Growth Factors, 20, 155-175.
http://dx.doi.org/10.1080/0897719031000084355 [5] Good, D., Polverini, P., Rastinejad, F., Le Beau, M., Lemons, R., Frazier, W. and Bouck, N. (1990) A Tumor Suppressor-Dependent Inhibitor of Angiogenesis Is Immunologically and Functionally Indistinguishable from a Fragment of Thrombospondin. Proceedings of the National Academy of Sciences, 87, 6624-6628.
http://dx.doi.org/10.1073/pnas.87.17.6624 [6] Bickel, S.T., Juliano, J.D. and Nagy, J.D. (2014) Evolution of Proliferation and the Angiogenic Switch in Tumors with High Clonal Diversity. PLoS ONE, 9, Article ID: e91992. [7] Folkman, J. (1971) Fundamental Concept of the Angiogenic Process. The New England Journal of Medicine, 285, 1182-1186. [8] Harrington, H.A., Maier, M., Naidoo, L., Whitaker, N. and Kevrekidis, P.G. (2006) A Hybrid Model for Tumor-Induced Angiogenesis in the Cornea in the Presence of Inhibitors. Mathematical and Computer Modelling, 46, 513-524.
http://dx.doi.org/10.1016/j.mcm.2006.11.034 [9] Tong, S. and Yuan, F. (2001) Numerical Simualtion of Angiogeneis in the Cornea. Microvascular Research, 61, 14-27. [10] Abdulle, A. (2002) Fourth Order Chebyshev Methods with Recurrence Relation. SIAM Journal on Scientific Computing, 23, 2041-2054. http://dx.doi.org/10.1137/S1064827500379549 [11] Van der Houwen, S.B, (1980) On the Internal Stage Runge-Kutta Methods for Large m-Values. Zeitschrift für Angewandte Mathematik und Mechanik, 60, 479-485. [12] Verwer, J.G. (1996) Explicit Runge-Kutta Methods for Parabolic Differential Equations. Applied Numerical Mathematics, 22, 359-379. http://dx.doi.org/10.1016/S0168-9274(96)00022-0 [13] Sommeijer, B., Shampine, L. and Verwer, J. (1998) RKC: An Explicit Solver for Parabolic PDEs. Journal of Computational and Applied Mathematics, 88, 326. [14] Riha, W. (1972) Optimal Stability Polynominals. Computing, 9, 37-43.
http://dx.doi.org/10.1007/BF02236374 [15] Abdulle, A. (2011) Explicit Stabilized Runge-Kutta Methods. MATHICSE Technical Report, Nr. 27.2011. [16] LeVeque, R.J. (2007) Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 175-179. http://dx.doi.org/10.1007/BF02236374