Physics-Informed Neural Networks for Chemotherapy Pharmacokinetics: Benchmarking the Clinical Estimator and Exposing Parameter Identifiability

Technical Summary: Physics-Informed Neural Networks for Chemotherapy Pharmacokinetics

Problem Statement
Chemotherapy dosing relies on understanding drug concentration over time in specific tissues, yet clinical practice is limited to measuring plasma concentrations via blood sampling. Tissue and tumor concentrations, which determine efficacy and toxicity, remain unobserved. Compartmental pharmacokinetic (PK) models are used to infer these hidden states. The standard clinical approach employs Nonlinear Least-Squares (NLS) fitting on an analytical biexponential solution derived from linear two-compartment Ordinary Differential Equations (ODEs). While effective under ideal assumptions (linear kinetics, low noise), this method fails when assumptions are violated (e.g., saturable elimination) or when the analytical solution does not exist. The paper investigates whether Physics-Informed Neural Networks (PINNs) can serve as a robust alternative that handles partial observations, nonlinear dynamics, and structural extensions without requiring closed-form solutions.

Methodology
The study benchmarks three approaches on two PK problems: a linear two-compartment model and a Michaelis–Menten extension (saturable elimination).

1. Baselines:
   • NLS: The clinical standard using scipy.optimize.curve_fit on the analytical biexponential plasma function. It fits parameters $(A, B, \alpha, \beta)$ and inverts them to rate constants, then integrates the ODE to estimate tissue concentration.
   • MLP (Data-Only): A Multi-Layer Perceptron with identical architecture to the PINN but trained solely on plasma data without physics constraints.
2. PINN Formulation:
   • A neural network $f_\theta$ maps time to plasma ($\hat{C}_p$) and tissue ($\hat{C}_t$) concentrations.
   • The loss function combines data fidelity ($L_{data}$), initial condition enforcement ($L_{ic}$), and the ODE residual ($L_{phys}$) calculated via automatic differentiation.
   • Parameters are stored as log-values to ensure positivity.
   • Optimization: A two-stage process using Adam for global exploration followed by L-BFGS for local refinement.
3. Experimental Setup:
   • Synthetic ground truth is generated for doxorubicin parameters over 24 hours.
   • Experiments include a headline run (8 samples, 10% noise), noise sweeps, sparsity sweeps, and a structural misspecification test where data is generated by the Michaelis–Menten ODE (no closed form).
   • Comparisons are performed across five random seeds to ensure statistical robustness.

Key Results

• Linear Two-Compartment Model:

  • Performance: NLS and PINN are statistically tied on plasma RMSE. NLS recovers elimination rate $k_e$ with slightly less bias, while the PINN is marginally biased high.
  • Hidden State Recovery: The PINN successfully recovers the tissue curve in a single training pass. In contrast, the data-only MLP fails to reconstruct the tissue compartment, exhibiting an RMSE roughly 10 times higher than the physics-based methods.
  • Efficiency: NLS is significantly faster (0.004s) compared to the PINN (~25s), reflecting the computational cost of the neural approach versus the analytical inversion.
  • Identifiability: Both methods struggle with the parameter $k_{12}$ due to practical non-identifiability (poor conditioning) in noisy data, a property of the data structure rather than the estimation method.

• Michaelis–Menten Extension (Saturable Elimination):

  • NLS Failure: Because the biexponential closed form does not exist for this system, the NLS estimator is mis-specified. It returns meaningless rate constants, silently failing to capture the true dynamics.
  • PINN Diagnostic: The PINN does not force a fit to a non-existent closed form. Instead, it converges to a basin where $k_{12} \to 0$, honestly exposing the structural non-identifiability of the model when relying solely on plasma data.
  • Partial Observability Resolution: Adding sparse tissue observations to the PINN loss largely resolves identifiability. Across five seeds, the PINN recovers $k_{21}$ within 1% of the truth and $V_{max}, K_m$ within one standard deviation. While $k_{12}$ remains biased (moving in the correct direction but $\sim 2\sigma$ below truth), this recovery is impossible for the NLS estimator, which cannot attempt the fit at all without a closed-form solution.

Significance and Claims
The paper explicitly states that its claim is not that PINNs outperform NLS in terms of raw accuracy or speed on well-specified, linear problems. Instead, the significance lies in the uniformity and diagnostic capability of the PINN approach:

1. Unified Recipe: PINNs provide a single framework that handles the textbook linear problem (matching NLS performance) and seamlessly extends to nonlinear, structural variations (like Michaelis–Menten) without requiring new analytical derivations.
2. Exposing Identifiability: Unlike the NLS estimator, which hides structural non-identifiability behind "silent" failure and meaningless parameter estimates, the PINN reveals these limitations directly. By converging to specific basins (e.g., $k_{12} \to 0$), the PINN acts as a diagnostic tool that signals when the data is insufficient to constrain the model.
3. Heterogeneous Data Integration: The PINN framework naturally absorbs heterogeneous measurements (plasma and sparse tissue data) within a single loss function, enabling the recovery of hidden compartments that data-only models fail to reconstruct.

In summary, the work presents a careful, multi-seed benchmark demonstrating that while PINNs may not replace the speed of analytical estimators in ideal scenarios, they offer a robust, extensible, and honest framework for complex PK problems where analytical solutions are unavailable or structural identifiability is a concern.