Learning Model Parameter Dynamics in a Combination Therapy for Bladder Cancer from Sparse Biological Data

In a mathematical model of interacting biological organisms, where external interventions may alter behavior over time, traditional models that assume fixed parameters usually do not capture the evolving dynamics. In oncology, this is further exacerbated by the fact that experimental data are often sparse and sometimes are composed of a few time points of tumor volume. In this paper, we propose to learn time-varying interactions between cells, such as those of bladder cancer tumors and immune cells, and their response to a combination of anticancer treatments in a limited data scenario. We employ the physics-informed neural network (PINN) approach to predict possible subpopulation trajectories at time points where no observed data are available. We demonstrate that our approach is consistent with the biological explanation of subpopulation trajectories. Our method provides a framework for learning evolving interactions among biological organisms when external interventions are applied to their environment.

💡 Research Summary

This paper addresses the challenge of modeling dynamic interactions among cancer cells, immune cells, suppressor cells, and chemotherapeutic agents in bladder cancer when experimental observations are extremely sparse. The authors focus on a pre‑clinical mouse model in which gemcitabine (GEM) is administered intravesically on day 10 to deplete myeloid‑derived suppressor cells (MDSCs), followed by adoptive transfer of OT‑1 T cells on day 14 to target MB49‑OVA tumors. Tumor volume is measured by ultrasound on six irregular days (6, 9, 13, 16, 20, 23), while flow cytometry provides the initial proportions of cancer cells (C) and MDSCs (M) at day 6, and digital histology supplies spatial counts of C, M, and T cells at days 17 and 23.

The biological system is formalized as a set of four ordinary differential equations (ODEs) describing the time evolution of cancer cells (C), OT‑1 T cells (T), MDSCs (M), and gemcitabine concentration (G). All parameters are assumed constant except for the MDSC‑mediated suppression rate of T cells, denoted s_MT(t), which is hypothesized to vary in response to the combined therapy. Traditional fixed‑parameter ODE fitting cannot capture such temporal variability, especially given the limited data.

To overcome this, the authors employ Physics‑Informed Neural Networks (PINNs). Two neural surrogates are constructed: u_NN(t) approximates the state vector (C, T, M, G), and Λ_NN(t) approximates the time‑varying parameter s_MT(t). The input to both networks is time, enriched by spline‑based interpolation that creates additional synthetic data points between the observed measurement times, thereby mitigating sparsity. The hidden layers use the SiLU activation function, while Softplus is applied at the output to enforce non‑negativity of concentrations and rates.

Training minimizes a composite loss L_total = w_r L_r + w_d L_d + w_IC L_IC + w_bc L_bc. L_r penalizes the ODE residuals computed via automatic differentiation; L_d measures discrepancy between u_NN and the interpolated tumor‑volume data; L_IC enforces the known initial condition at day 6; L_bc incorporates domain‑specific constraints derived from flow cytometry (C and M proportions) and histology (cell counts at days 17 and 23). The weights w_* are adaptively tuned using a multi‑task learning scheme that balances the contributions of each term. Training proceeds for 20 000 epochs with the Adam optimizer, using three hidden layers of 100 neurons for u_NN and two hidden layers of 200 neurons for Λ_NN, all implemented in PyTorch.

Results show that the PINN successfully reconstructs plausible trajectories for each subpopulation, matching the observed total tumor volume (C + T + M) at the six measurement days. The learned s_MT(t) exhibits a sharp decline following GEM injection (reflecting MDSC depletion) and a subsequent rise after OT‑1 T‑cell infusion (reflecting restored immune suppression). These dynamics align with biological expectations: GEM reduces MDSC numbers, thereby temporarily relieving T‑cell inhibition; later, the expanding T‑cell population re‑establishes regulatory feedback. The model also captures gemcitabine pharmacokinetics, showing rapid clearance after the single dose. To assess uncertainty, the authors repeat the training ten times with different random seeds, producing confidence bands around each trajectory; the bands remain narrow, indicating robustness despite data scarcity.

In the discussion, the authors highlight three major contributions. First, they demonstrate that PINNs can infer time‑varying parameters from highly limited, irregularly sampled data, a scenario common in oncology and other biomedical fields. Second, by embedding explicit biological constraints into the loss function, the approach ensures that learned solutions respect known physiology, reducing the space of admissible solutions and preventing overfitting to noise. Third, the framework is modular and can be extended to other cancers, additional therapeutic agents, or even non‑oncologic systems such as infectious disease models where interventions alter interaction rates over time.

Funding acknowledgments note support from the U.S. Department of Defense (grant W81XWH‑22‑1‑0340) and the National Cancer Institute (R01‑CA259387, P30‑CA076292), as well as a high‑school internship program (U54 CA274507‑02A1). The paper concludes that integrating mechanistic ODE models with data‑driven PINNs offers a powerful tool for extracting hidden dynamical information from sparse biomedical datasets, paving the way for more informed treatment design and personalized therapy planning.