Three-dimensional modeling of tumor angiogenesis: endothelial cell progression and therapeutic implications

Abstract

We propose a three-dimensional mathematical model describing the dynamics of endothelial cells during tumor-induced angiogenesis. This framework extends our earlier one- and two-dimensional formulations by incorporating the three-dimensional spatial dynamics of endothelial cell migration, proliferation, and interaction with the extracellular matrix. The model is coupled to the corresponding one- and two-dimensional systems through transmission and boundary conditions that describe the migration of endothelial cells from the vessel wall into the extracellular matrix and their subsequent progression toward the tumor region. It accounts for key biophysical and biochemical mechanisms involved in angiogenesis, including protease-mediated matrix degradation, fibronectin remodeling, and the competing effects of pro-angiogenic and anti-angiogenic factors such as vascular endothelial growth factor and angiostatin. We present numerical simulations using Matlab, which exhibit strong qualitative agreement with experimental observations, capturing the onset of tumor vascularization and the accelerated progression of capillary tip growth as the sprout approaches the tumor region. In addition, the simulations reveal the formation of fibronectin channels and demonstrate the inhibitory impact of angiostatin on endothelial cell migration and matrix remodeling. We also compute the speed of the migration of endothelial cells as they pass across the extracellular matrix, showing that the endothelial cells move faster as they approach the tumor source. Overall, the proposed three-dimensional framework provides a biologically realistic and computationally robust tool for investigating tumor-driven angiogenesis and anti-angiogenic therapeutic strategies.