Multiscale angiogenesis modeling using mixed finite element methods

In this paper, we present a deterministic two-scale tissue-cellular approach for modeling growth factor-induced angiogenesis. The bioreaction-diffusion of chemotactic growth factors (CGF) is modeled at a tissue scale, whereas cell proliferation, capillary extension, branching, and anastomosis are modeled at a cellular scale. The capillary indicator function is used to bridge these two scales. The complete system of equations consists of parabolic PDEs coupled nonlinearly with a varying number of ODEs and algebraic equations. Our proposed schemes involve applying mixed finite element methods to approximate concentrations of CGF and a point-to-point tracking method to simulate sprout branching and anastomosis. Capillary extensions are computed by a system of ODEs. Here, both the continuous and discrete-in-time algorithms are analyzed using some new techniques for treating the nonlinear coupling terms. 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 are established. Optimal order estimates in the mesh size are obtained for the continuous-in-time schemes, and optimal order estimates in both the mesh and the time step sizes are derived for the fully discretized schemes. In addition, we address several implementation issues, including an equivalent cell-centered finite difference formulation, time splitting techniques, and object-oriented programming strategies for efficient scientific computing. Finally, a representative simulation example is provided.