On a Mathematical Model Describing Chemotherapeutic Drug Treatment for Tumor Cells
Xiaoqin Liu, Hong-Ming Yin
TL;DR
We study a four-species reaction-diffusion-advection PDE model for normal tissue $N$, tumor $T$, immune $I$, and drug $U$, incorporating strong Allee effects to capture extinction thresholds. The analysis proves global well-posedness of nonnegative weak solutions under mild structural assumptions and develops a CNBE numerical scheme to simulate pulsed chemotherapy in heterogeneous tissue. Numerical experiments reveal a scheduling trade-off: concentrated high-rate pulses achieve rapid early tumor knock-down, while more frequent, gentler pulses better preserve normal tissue with comparable control of the invasive front over four weeks. The results provide a mathematically rigorous framework and practical guidance for spatially targeted chemotherapy regimen design.
Abstract
In this paper, we study a semilinear parabolic PDE system which describes the interaction of normal cells, tumor cells, immune cells, with a chemotherapeutic drug. The model extends the previous model with incorporating strong Allee affects in the normal-tissue and tumor dynamics. Under mild assumptions, we establish global-in-time existence and uniqueness of nonnegative weak solutions and derive L-infinity bounds for all time. We then investigate spatiotemporal dynamics of the model and therapy scheduling using an implicit Crank Nicolson Backward Euler (CNBE) scheme. Simulations in a heterogeneous two-dimensional space-dimensional tissue region with three tumor peaks indicate rapid tumor invasion without treatment and significant tumor suppression under pulsed chemotherapeutic treatment. Moreover, in a fixed total dose delivered within the treatment cycle, while keeping each injection duration fixed, concentrated pulses produce stronger early knock-down of tumor density, while more frequent but gentler pulses achieve comparable control of the tumor invasive front while better preserving normal tissue over a four-week period.