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On a Mathematical Model Arising from an Optimal Chemotherapeutic Drug Treatment for Tumor Cells
Dissertation

On a Mathematical Model Arising from an Optimal Chemotherapeutic Drug Treatment for Tumor Cells

Xiaoqin (Kurt) Liu
Doctor of Philosophy (PhD), Washington State University
2026
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Open Access CC BY V4.0

Abstract

We develop and analyze a four-species reaction-diffusion-advection model for normal tissue ($N$), tumor ($T$), immune effector cells ($I$), and chemotherapeutic drug ($U$). The model extends previous tumor--immune--drug frameworks by incorporating Allee effects in the $N$ and $T$ equations and allowing general second-order elliptic operators. Under minimal structural assumptions, we prove local well-posedness of weak solutions via the Schauder fixed-point theorem. A blow-up alternative then promotes local to global-in-time existence and uniqueness once $L^\infty$ bounds are established. For spatiotemporal dynamics and treatment design, we employ a conservative Crank Nicolson/Backward Euler (CN-BE) scheme with homogeneous Neumann boundary conditions enforced via ghost points. Simulations on a heterogeneous two-dimensional tissue with three tumor peaks show that (i) without drug, tumors rapidly approach carrying capacity and form broad invasive fronts, while $I$ develops an annular band advancing inward toward tumor foci yet remains insufficient to halt tumor growth; and (ii) with chemotherapeutic treatment, pulsed dosing significantly suppresses tumor burden. For a fixed per-pulse input $V_0\tau$, varying amplitude $V_0$, duration $\tau$, and number of injections $N^*$ reveals dose-time-frequency trade-offs. The analysis provides a rigorous foundation for the model, and the simulations offer a practical tool for exploring chemotherapeutic scheduling in spatially heterogeneous tumors.

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