Solutions of a Linearized Mathematical Model for Capillary Formation in Tumor Angiogenesis: an Initial Data Perturbation Approximation

Overview

Journal Comput Math Methods Med

Publisher Hindawi

Specialty Medical Informatics

Date 2013 Sep 13

PMID 24027602

Authors

Serdal Pamuk

Affiliations

Soon will be listed here.

Abstract

We present a mathematical model for capillary formation in tumor angiogenesis and solve it by linearizing it using an initial data perturbation method. This method is highly effective to obtain solutions of nonlinear coupled differential equations. We also provide a specific example resulting, that even a few terms of the obtained series solutions are enough to have an idea for the endothelial cell movement in a capillary. MATLAB-generated figures are provided, and the stability criteria are determined for the steady-state solution of the cell equation.

References

Folkman J . The vascularization of tumors. Sci Am. 1976; 234(5):58-64, 70-3. DOI: 10.1038/scientificamerican0576-58. View

Yamada K, Olden K . Fibronectins--adhesive glycoproteins of cell surface and blood. Nature. 1978; 275(5677):179-84. DOI: 10.1038/275179a0. View

LEVINE H, Sleeman B, Nilsen-Hamilton M . Mathematical modeling of the onset of capillary formation initiating angiogenesis. J Math Biol. 2001; 42(3):195-238. DOI: 10.1007/s002850000037. View

Pamuk S . Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source. Math Biosci. 2004; 189(1):21-38. DOI: 10.1016/j.mbs.2004.01.006. View

LEVINE H, Pamuk S, Sleeman B, Nilsen-Hamilton M . Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull Math Biol. 2001; 63(5):801-63. DOI: 10.1006/bulm.2001.0240. View

Schor A, Schor S . Tumour angiogenesis. J Pathol. 1983; 141(3):385-413. DOI: 10.1002/path.1711410315. View

LEVINE H, Sleeman B, Nilsen-Hamilton M . 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 Biosci. 2000; 168(1):77-115. DOI: 10.1016/s0025-5564(00)00034-1. View

Folkman J . Tumor angiogenesis: therapeutic implications. N Engl J Med. 1971; 285(21):1182-6. DOI: 10.1056/NEJM197111182852108. View

Nicosia R, Bonanno E, Smith M . Fibronectin promotes the elongation of microvessels during angiogenesis in vitro. J Cell Physiol. 1993; 154(3):654-61. DOI: 10.1002/jcp.1041540325. View

10.

PAWELETZ N, Knierim M . Tumor-related angiogenesis. Crit Rev Oncol Hematol. 1989; 9(3):197-242. DOI: 10.1016/s1040-8428(89)80002-2. View

11.

Verbeni M, Sanchez O, Mollica E, Siegl-Cachedenier I, Carleton A, Guerrero I . Morphogenetic action through flux-limited spreading. Phys Life Rev. 2013; 10(4):457-75. DOI: 10.1016/j.plrev.2013.06.004. View

12.

Schleef R, Birdwell C . The effect of proteases on endothelial cell migration in vitro. Exp Cell Res. 1982; 141(2):503-8. DOI: 10.1016/0014-4827(82)90244-0. View