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Mathematical Modeling of Cancer-Bacterial Therapy: Analysis and Numerical Simulation via Physics-Informed Neural Networks

Bacterial cancer therapy exploits anaerobic bacteria's ability to target hypoxia tumor regions, yet the interactions among tumor growth, bacterial colonization, oxygen levels, immunosuppressive cytokines, and bacterial communication remain poorly quantified. We present a mathematical model of five coupled nonlinear reaction-diffusion equations in a two-dimensional tissue domain. We proved the global well-posedness of the model and identified its steady states to analyze stability. Furthermore, a physics-informed neural network (PINN) solves the system without a mesh and without requiring extensive data. It provides convergence guarantees by combining residual stability and Sobolev approximation error bounds. This results in an overall error rate of O(n^-2 ln^4(n) + N^-1/2), which depends on the network width n and the number of collocation points N. We conducted several numerical experiments, including predicting the tumor's response to therapy. We also performed a sensitivity analysis of certain parameters. The results suggest that long-term therapeutic efficacy may require the maintenance of hypoxia regions in the tumor, or using bacteria that tolerate oxygen better, may be necessary for long-lasting tumor control.

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