Multiscale angiogenesis modeling

We propose a deterministic two-scale tissue-cellular approach for modeling growth factor-induced angiogenesis. The bioreaction and diffusion of capillary growth factors (CGF) are modeled at a tissue scale, whereas capillary extension, branching and anastomosis are modeled at a cellular scale. The capillary indicator function is used to bridge these two scales. To solve the equation system numerically, we construct a two-grid algorithm that involves applying a mixed finite element method to approximate concentrations of CGF on a coarse mesh and a point-to-point tracking method to simulate sprout branching and anastomosis on a fine grid. An analysis of the algorithm establishes optimal error bounds for each of the processes - CGF reaction-diffusion, capillary extension, sprout branching and anastomosis - and overall error bounds for their coupled nonlinear interactions.