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

Ayoub Farkane; David Lassounon

q-bio.QMMar 18, 2026

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

Authors: Ayoub Farkane, David Lassounon

Organizations: TICLab, International University of Rabat, Rabat, 11100, Morocco · Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France · Université de Lorraine, CNRS, IECL, F-54000, Nancy, France · Univ Rennes, INSA, CNRS, IRMAR-UMR 6625, F-35000, 35708, Rennes, France

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.

Figures & tables

Figure 1: Simplified diagram of the main interactions variables

\displaystyle T,B,O,I,S

Equation	Description	Symbol	Value	Unit
All	Tumor cell motility	$\displaystyle D_{T}$	$\displaystyle 0.01$	mm 2 day -1
	Bacterial diffusivity	$\displaystyle D_{B}$	$\displaystyle 0.1$	mm 2 day -1
	Oxygen diffusivity	$\displaystyle D_{O}$	$\displaystyle 1.0$	mm 2 day -1
	Cytokine diffusivity	$\displaystyle D_{I}$	$\displaystyle 0.5$	mm 2 day -1
	AHL signal diffusivity	$\displaystyle D_{S}$	$\displaystyle 0.3$	mm 2 day -1
(1) Tumor $\displaystyle T$	Proliferation rate	$\displaystyle\rho_{T}$	$\displaystyle 0.3$	day -1

Table 1: The numerical simulation parameters of the system ( 8 )

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.

Symbol	Color	Location $\displaystyle(x,y)$	Purpose
$\displaystyle\color[rgb]{1,0,0}{\times}$	Red	$\displaystyle(3.0,3.0)$	Tumor center-initial tumor seed location
$\displaystyle\color[rgb]{0,0,1}{\times}$	Blue	$\displaystyle(1.8,3.6)$	Bacteria injection site, 30% $\displaystyle x$ , 60% $\displaystyle y$ of domain
$\displaystyle\color[rgb]{0,1,0}{\times}$	Green	$\displaystyle(3.0,5.5)$	Top edge, Near-boundary monitoring point
$\displaystyle\color[rgb]{1,0.5,0}{\times}$	Orange	$\displaystyle(3.0,0.5)$	Bottom edge, Near-boundary monitoring point
$\displaystyle\color[rgb]{0.55,0.14,1}{\times}$	Purple	$\displaystyle(5.5,3.0)$	Right edge, Near-boundary monitoring point

Table 2: Monitoring Probe Locations in Figure 1

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.

Figure 6: Oxygen deplation $\displaystyle\&$ hupoxia dynamics, $\displaystyle O_{2}$ consumed by tumor.

Figure 7: Immune cytokine response (I). Cytokines are produced by tumor cells and regulate immune activity.

Figure 8: Bacterial quorum-sensing signal (S) dynamics.

Figure 9: All 5 variables at fixed $\displaystyle(x=3.0,y=3.0)$ : solid lines represent (T, B, O normalized), dashed lines represent (I, S normalized).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 10 — Figure 10: Sensitivity of the model to the specific parameters $\displaystyle\alpha_{S}$ , $\displaystyle\beta_{I}$ , and $\displaystyle\gamma_{vas}$ .

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William Lavery, Jodie A. Cochrane, Christian Olesen, Dagim S. Tadele+2

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Dynamical Systems Reaction-Diffusion Systems PDE Discovery Physics-Informed ML

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Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to

1\mathrm{D}{+}t

reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to

2\mathrm{D}{+}t

systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering

2\mathrm{D}{+}t

reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.

Abstract

Reaction-Diffusion Systems

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