Two-dimensional mathematical modeling of tumor invasion using the generalized integral transform technique: Role of haptotaxis in spatial tumor-ECM dynamics.

Abstract

e15005 Background: Tumor invasion is a spatially heterogeneous and multiscale process driven by interactions between cancer cells, the extracellular matrix (ECM), and matrix degrading enzymes (MDEs). Mathematical modeling is essential for investigating invasion mechanisms that are difficult to assess experimentally. Directed migration, particularly haptotaxis along gradients of matrix bound ECM components, is a key determinant of invasion morphology and aggressiveness. While one dimensional models provide insight, two dimensional (2D) formulations are required to capture realistic spatial invasion patterns. However, 2D models present significant numerical challenges due to strong nonlinear coupling and steep gradients, motivating advanced computational approaches such as the Generalized Integral Transform Technique (GITT). Methods: A dimensionless 2D system of coupled partial differential equations was developed to describe tumor cell density, ECM density, and MDE concentration within a confined tissue domain. Tumor cell migration combined random motility and haptotaxis toward ECM gradients. ECM degradation was locally driven by enzymatic activity, while MDE dynamics included diffusion, tumor induced production, and decay. Homogeneous Neumann boundary conditions represented a closed system. The system was solved using GITT, transforming the PDEs into a coupled set of ordinary differential equations via eigenfunction expansions in both spatial directions. The resulting stiff ODE system was integrated using the IMSL DIVPAG routine. Convergence with respect to truncation order was assessed, and numerical verification was performed through comparison with the Method of Lines. Parametric analyses focused on haptotactic sensitivity. Results: The proposed approach produced stable and convergent solutions for all state variables, with excellent agreement relative to reference numerical solutions. Low haptotactic sensitivity (γ = 0.0025) resulted in diffuse and nearly isotropic tumor expansion dominated by random motility. Increasing haptotactic sensitivity (γ = 0.01) led to strongly localized and anisotropic invasion patterns, characterized by sharp fronts, spatial channeling, and intensified ECM degradation. Despite increased nonlinearity, GITT maintained accuracy provided sufficiently high truncation orders were employed. Conclusions: The results demonstrate that GITT is a robust and efficient framework for two dimensional tumor invasion modeling. Haptotaxis emerged as a key regulator of invasion aggressiveness and spatial heterogeneity, shaping tumor morphology and ECM remodeling. This approach provides mechanistic insight into ECM guided migration and supports future in silico studies of tumor microenvironment interactions and anti invasive strategies.