A Stabilized Numerical Framework for Necrotic Tumor Growth via Coupled Boundary Integral and Obstacle Solvers

TL;DR

This paper introduces a stable numerical method for simulating tumor growth with necrotic cores, effectively handling complex boundary evolution and obstacle problems through a coupled predictor-corrector scheme.

Contribution

A novel stabilized computational framework combining boundary integral and obstacle solvers for accurate tumor growth modeling with necrosis.

Findings

Robust simulation of necrotic tumor growth with complex geometries.

Convergence theory established for single-interface cases.

Effective capture of necrotic core nucleation and topological changes.

Abstract

We present a robust computational framework for Hele-Shaw tumor growth with necrotic cores, a problem identified as the incompressible limit of the Porous Media Equation. Simulating this system presents a fundamental challenge: while the outer boundary evolves via advection, the inner necrotic interface is defined by an obstacle problem and lacks an explicit advection structure, causing standard schemes to fail. To address this, we introduce a stabilized predictor-corrector strategy that iteratively resolves the bidirectional coupling between the nutrient-pressure fields and the domain geometry, ensuring robust time-stepping for both the advection-driven outer surface and the obstacle-defined necrotic core. We establish rigorous convergence theory for the single-interface case and demonstrate the method's robustness in capturing the topological transition of necrotic core nucleation and complex geometric evolution.