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

Ayoub Farkane, David Lassounon

Abstract

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.

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

Abstract

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.
Paper Structure (39 sections, 11 theorems, 117 equations, 10 figures, 2 tables, 1 algorithm)

This paper contains 39 sections, 11 theorems, 117 equations, 10 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Motivation of PINNs
  3. Biological Context of Bacterial Cancer Therapy
  4. Existing Mathematical Modeling
  5. Deep Learning Approaches to Tumor Modeling
  6. Contributions
  7. Mathematical Modeling
  8. Evolution of Tumor Density under Treatment
  9. Evolution of Bacterial Density
  10. Evolution of Oxygen Density
  11. Evolution of Immunosuppressive Cytokines Density
  12. Evolution of Diffusible Bacterial Signals
  13. Mathematical Analysis
  14. Existence and Uniqueness of Global Classical Solutions
  15. Nondimensionalization of the system
  16. ...and 24 more sections

Key Result

Lemma 3.1

The nonlinearity $\mathrm{F}=(F_{T},F_{B},F_{O},F_{I},F_{S})^{\top}$ from couple with $U=(T,B,O,I,S)^{ \intercal}\ge0$ componentwise and is quasi-positive and satisfies the mass-control structure.

Figures (10)

  • Figure 1: Simplified diagram of the main interactions variables $T,B,O,I,S$.
  • Figure 2: Training and validation loss evolution over 8000 epochs.
  • Figure 3: Spatio-temporal evolution of the five-species tumor-bacteria system over 30 days.
  • Figure 4: Tumor cell dynamics: growth phase, peak and bacterial-driven degradation
  • Figure 5: Bacterial therapy effect: Salmonella-tumor interaction dynamics bacteria colonise hypoxia régions, produce signals, and drive tumors regression.
  • ...and 5 more figures

Theorems & Definitions (28)

  • Definition 3.1
  • Lemma 3.1
  • Proof 1
  • Theorem 3.1
  • Proof 2
  • Proposition 3.1
  • Proof 3
  • Remark 3.1
  • Proposition 3.2
  • Proof 4
  • ...and 18 more