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On the qualitative analysis of the uniqueness of the movement of endothelial cells

This paper extends the work of Pamuk (2003) by showing mathematically that the movement of endothelial cells, to the regions where active enzyme is large or where fibronectin is small, is unique. To do this, we obtain the existence and uniqueness of the steady-state solution of an initial-boundary value problem which mathematically models endothelial cell movement in tumor angiogenesis. | Turk J Math 34 (2010) , 367 – 375. ¨ ITAK ˙ c TUB doi:10.3906/mat-0903-26 On the qualitative analysis of the uniqueness of the movement of endothelial cells Erdem Altunta¸c and Serdal Pamuk Abstract This paper extends the work of Pamuk (2003) by showing mathematically that the movement of endothelial cells, to the regions where active enzyme is large or where fibronectin is small, is unique. To do this, we obtain the existence and uniqueness of the steady-state solution of an initial-boundary value problem which mathematically models endothelial cell movement in tumor angiogenesis. A specific example showing the instability of this steady-state solution is provided. Key Words: Existence; uniqueness; tumor angiogenesis; steady-state solution; transition probability density function. 1. Introduction Angiogenesis is a morphogenic process whereby new blood vessels are induced to grow out of a pre-existing vasculature. It is essential for tumor progression and is critical for the growth of primary cancer. It is well known that it occurs in three sequential steps [13]. First, the endothelial cell (EC) lining the vascular basal lamina (BL) (or basement membrane) degrade this membrane. Second, the EC migrate and proliferate (via mitosis) into the extra cellular matrix (ECM). Finally, capillary loops form. In recent years, progress has been made to understand this phenomenon at the molecular level. This includes the identification of potent angiogenic factors, the discovery of the role of proteases, the importance of the ECM, and the emerging characterization of signal transduction pathways in EC. One of the major components of the ECM is fibronectin, a large, highly adhesive glycoprotein particularly abundant in plasma, connective tissue matrices, and BL [8]. It is also known to enhance EC adhesion to collagen and is produced by EC. The simplest unifying interpretation of these findings is that it functions as adhesive proteins that bind cells to other cells or to a .