Speaker
Description
We present a mathematical and computational framework for the numerical simulation of tumor-induced angiogenesis, a process of central relevance in sustainable healthcare modeling and cancer treatment planning. The continuous model consists of a five-component PDE system coupling endothelial cell density $C$, protease concentration $P$, inhibitor concentration $I$, extracellular matrix density $F$, and oxygen concentration $O$, where oxygen acts as a key regulator of vascular growth through Michaelis-Menten kinetics. Existence, uniqueness, boundedness of solutions, and the existence of a global attractor are established analytically (De Luca and Marcellino, 2025).
For the numerical solution, we develop a conservative implicit-explicit (IMEX) Modified Patankar method that overcomes a fundamental limitation of standard discretisations: the violation of solution positivity near steep gradients or when concentrations approach zero. The scheme treats diffusion implicitly via Crank-Nicolson and handles chemotaxis and reaction terms through a Modified Patankar formulation built on a novel \emph{flux-based production-destruction decomposition}. Interfacial fluxes are uniquely defined at cell boundaries via upwinding, ensuring discrete mass conservation; both production and destruction terms are weighted by Patankar denominators, yielding a genuine Modified Patankar scheme in the sense of Burchard, Deleersnijder, and Meister (2003). Positivity preservation is proven under a mild diffusive CFL condition.
The resulting method is first-order in time and second-order in space. Numerical experiments on the five-component angiogenesis model confirm the theoretical predictions: the scheme preserves positivity without a single violation across all test scenarios, including stress tests with near-zero initial data and time steps exceeding the CFL limit by a factor of $50$. In practical simulations, the IMEX-Patankar method permits time steps up to two orders of magnitude larger than fully explicit Runge-Kutta integration, achieving a measured speed-up of approximately $26\times$ in the number of time steps on the reference angiogenesis benchmark with $M=256$ grid points.