Mathematical modeling of Gompertz cancer cell dynamics using real data-driven physics informed neural networks

Tumor growth is a complex biological process governed by cellular proliferation, nutrient availability, and environmental constraints. Classical growth laws such as the logistic and Montroll models have been widely used to describe tumor dynamics; however, they often fail to accurately capture the gradual deceleration observed during later growth stages. In this work, we develop a physics-informed neural network (PINN) framework based on the Gompertz growth model to investigate biologically constrained tumor-growth dynamics from experimental data. Unlike purely data-driven approaches, the proposed framework incorporates the governing differential equation directly into the neural-network training process, enabling simultaneous data fitting and mechanistic consistency. The model is validated using publicly available tumor spheroid volume data of Chinese hamster fibroblast cells. Numerical results demonstrate that the proposed approach accurately reproduces the observed tumor-growth trajectory while providing biologically interpretable parameter estimates. In addition to standard error metrics, the study discusses the role of physics-based regularization, parameter sensitivity, and limitations associated with small biological datasets. Although the Gompertz equation admits a closed-form analytical solution, the present work serves as a benchmark proof-of-concept for applying PINNs to biologically motivated growth models and provides a foundation for future investigations of more complex tumor systems where analytical solutions are unavailable.