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On a Cross-Diffusion PDE-ODE Hybrid Model Arising from Cancer Invasion
Dissertation

On a Cross-Diffusion PDE-ODE Hybrid Model Arising from Cancer Invasion

Guanjun Pan
Doctor of Philosophy (PhD), Washington State University
2026
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Embargoed Access, Embargo ends: 01/20/2027 CC BY V4.0

Abstract

Cancer Invasion Cross-diffusion system Heterogeneous pattern Plasminogen activation system Well-posedness
This dissertation studies a strongly coupled cross-diffusion hybrid model describing cancer invasion in the plasminogen activation system, explicitly incorporating interactions between cancer cells and host normal cells. The model extends a diffusion-taxis-reaction model by introducing mutual cross-diffusion mechanisms between cancer and normal cells, chemotaxis and haptotaxis effects, logistic growth, recruitment, and degradation, and additional biochemical components, including vitronectin, urokinase-type plasminogen activator, plasminogen activator inhibitor, and plasmin. The mathematical model consists of nonlinear parabolic equations in divergence form with mutual cross-diffusion and coupled with ordinary differential equations. Under proper structural assumptions on coefficients, several a priori estimates are established. By employing a decoupling strategy, we prove the global existence and uniqueness of a non-negative weak solution in spatial dimension d<=2. In the three-dimensional case d=3, we prove the global existence of a weak solution under an explicit condition on cross-diffusion coefficients. In addition to the theoretical analysis, numerical simulations of the model are performed in both one- and two-dimensional spatial domains. The numerical scheme combines the Crank–Nicolson method, an upwind-biased discretization, and an explicit method for diffusion, cross-diffusion, and reaction terms, respectively. The simulations illustrate the impact of cross-diffusion, various bounded functions $B(u)$, and parameters. In particular, the simulations reveal the emergence of heterogeneous spatial patterns and the role of cross-diffusion coefficients and the bounded function B(u) in regulating aggregation.

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