On the Estimation of the Threshold Constant to Initiate Angiogenesis in a 3D Mathematical Model

In this study, we investigate the condition necessary for basement membrane degradation within a three-dimensional mathematical model of tumor angiogenesis. By analyzing the steady-state solution under certain assumptions, we estimate the threshold constant that triggers the governing equations of the model. Once the concentration of the tumor angiogenesis factor reaches this threshold, endothelial cells begin to migrate into the extracellular matrix, initiating angiogenesis. Furthermore, our findings offer valuable insights into the mechanisms of cell migration, which play a critical role in tumor angiogenesis. We also present MATLAB-generated figures comparing the numerical and steady-state solutions, providing strong evidence for the stability of this steady-state.