Physics-informed neural networks enable patient-specific tumor microenvironment modeling: Identifying parameters in combined immune therapy with integer- and fractional-order dynamics

The tumor immune microenvironment (TIME) is a dynamic network of immune cells, cytokines such as Interleukin-10 (IL-10), and immunosuppressive pathways like Programmed Cell Death Protein 1/Programmed Death-Ligand 1 (PD-1/PD-L1). Combined therapy targeting PD-1/PD-L1 (via checkpoint inhibitors) and IL-10 (via neutralizing antibodies) may synergistically remodel TIME to enhance antitumor immunity ( Sardar et al., 2023 ). Understanding the underlying mechanisms and interactions within this system is critical for the advancement of immunotherapy design. Existing mathematical models for TIME face significant challenges in identifying patient-specific physiological parameters, which often range from dozens to hundreds. To address this, we propose a computational framework, TIME-Net, based on physics-informed neural networks (PINNs) and Bayesian PINNs. This framework not only solves forward and inverse TIME models, but also identifies all 23 unknown parameters across 8 equations. The key innovations of TIME-Net include: (1) adopting active learning to reduce training data requirements, (2) constraining parameter search ranges to minimize correlations, (3) employing mixed-data augmentation to improve accuracy for low-sensitivity parameters, and (4) clustering parameters by sensitivity for group-wise identification. We applied TIME-Net to both integer- and fractional-order TIME models, evaluating three evolutionary scenarios: immune escape, tumor-immune coexistence, and tumor-free. Our results show that TIME-Net achieves high accuracy in parameter inference, with maximum relative errors of 2% (integer-order) and 8% (fractional-order). The framework remains robust under sparse, noisy (e.g., 5% noise in tumor-only data), and heterogeneous data conditions. Validation using synthetic data (simulating idealized scenarios) and in vitro experimental data confirmed its reliability. Comprehensive sensitivity and identifiability analyses, in conjunction with ablation studies, underscored the necessity of each technical component. Furthermore, we quantified computational uncertainty to demonstrate the robustness of TIME-Net. Finally, by optimizing drug doses for combined immunotherapy, we illustrate its translational potential. This study highlights the ability of TIME-Net to simultaneously identify dozens of patient-specific parameters and its promise for clinical adoption in personalized cancer treatment. • We proposed a computational framework for parameters identification of TIME models. • We quantify the uncertainty of the computational results of TIME models. • We fit the TIME modes to the in-vitro data. • We demonstrate the effectiveness of the proposed computational framework for parameter identification.