Drug Release Modeling using Physics-Informed Neural Networks

TL;DR

This work embedded Fick's second law into PINN as loss with 10,000 Latin-hypercube collocation points and utilized previously published experimental datasets to assess drug release performance through mean absolute error (MAE) and root mean square error (RMSE), considering noisy conditions and limited-data scenarios.

Abstract

Accurate modeling of drug release is essential for designing and developing controlled-release systems. Classical models (Fick, Higuchi, Peppas) rely on simplifying assumptions that limit their accuracy in complex geometries and release mechanisms. Here, we propose a novel approach using Physics-Informed Neural Networks (PINNs) and Bayesian PINNs (BPINNs) for predicting release from planar, 1D-wrinkled, and 2D-crumpled films. This approach uniquely integrates Fick's diffusion law with limited experimental data to enable accurate long-term predictions from short-term measurements, and is systematically benchmarked against classical drug release models. We embedded Fick's second law into PINN as loss with 10,000 Latin-hypercube collocation points and utilized previously published experimental datasets to assess drug release performance through mean absolute error (MAE) and root mean square error (RMSE), considering noisy conditions and limited-data scenarios. Our approach reduced mean error by up to 40% relative to classical baselines across all film types. The PINN formulation achieved RMSE <0.05 utilizing only the first 6% of the release time data (reducing 94% of release time required for the experiments) for the planar film. For wrinkled and crumpled films, the PINN reached RMSE <0.05 in 33% of the release time data. BPINNs provide tighter and more reliable uncertainty quantification under noise. By combining physical laws with experimental data, the proposed framework yields highly accurate long-term release predictions from short-term measurements, offering a practical route for accelerated characterization and more efficient early-stage drug release system formulation.

Figures (8)

• Figure 1: Reprinted (adapted) with permission from liu2021. Copyright 2021 American Chemical Society. The figure shows the molecular release mechanisms in different nanosheet films. The figure displays three types of films used in the controlled molecular release study: Planar, 1D wrinkled, and 2D crumpled. Each structure facilitates a unique release profile: Planar films allow a uniform release from the basal plane, while 1D wrinkled and 2D crumpled films enable more complex, controlled release, especially from the film edges, creating a heterogeneous release profile. The graphene oxide nanosheet structures serve as carriers for intercalant molecules, with release observed from both the basal plane and edges.
• Figure 2: Detailed PINN and BPINN architectures. (a) The standard PINN framework integrates a neural network with physical laws. Inputs, represented by concentration $c$ and time $t$, are fed into a fully-connected neural network with activation functions ($\sigma$) in each hidden layer. The network predicts $u$, which is subjected to physical constraints based on data driven loss and Fick’s law of diffusion: $f(x, t) = \frac{\partial u}{\partial t} - D \frac{\partial^2 u}{\partial x^2}$, where $D$ denotes the diffusion coefficient. This physical loss is computed by applying automatic differentiation to enforce the equation at collocation points. The output $f(x, t)$ is then used to calculate the total loss, combining both data-driven loss and physics-based loss. The training loop iterates until the total loss falls below a predefined threshold $\epsilon$, at which point the model is considered trained and outputs a “Done” signal. (b) The BPINN extends the PINN by incorporating Bayesian inference, allowing for uncertainty quantification in the model parameters. In this architecture, a Bayesian Neural Network (BNN) with inputs $c$ and $t$ generates probabilistic predictions of $u$, modeling each weight and bias as a distribution rather than a fixed value. The physical law, represented as the likelihood $P(D|\theta)$, is enforced by the physics-informed part of the network. Here, the system dynamics are governed by equations for $\tilde{f} = N_x(\tilde{u}; \lambda)$ and $\tilde{b} = B_x(\tilde{u}; \lambda)$, where $\tilde{u}$ represents predictions with uncertainty and $\lambda$ are additional parameters representing model priors. Observations $u$, $b$, and $f$ provide data-driven grounding. The Bayesian inference is applied through Bayes' rule: $P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)}$, where $P(\theta)$ is the prior derived from the BNN and $P(D|\theta)$ is the physics-informed likelihood. The posterior $P(\theta|D)$ is obtained via sampling techniques such as Hamiltonian Monte Carlo (HMC) or Variational Inference (VI), providing a distribution over the model parameters.
• Figure 3: Drug release data for flat, 1D, and 2D films fitted using classical models and PINNs. (Top left) The flat film data shows the release profile of drug concentration over normalized time. Each model attempts to capture the diffusion characteristics and release of therapeutics, with the PINN model closely aligning with experimental data points. (Top right) The 1D wrinkled film data, which adds a layer of structural complexity, shows a different release profile. Classical models fit the data, but the PINN model’s flexibility allows it to accommodate the nuanced release dynamics more effectively. (Bottom) The 2D crumpled film data represents the most complex structure, with multi-dimensional diffusion effects. The PINN model maintains a consistent fit across the release profile, capturing both rapid initial release and slower sustained release phases, while classical models demonstrate limitations. The total time (normalized time = 1) for all release curves is 48 hours.
• Figure 4: MAE and RMSE for classical models and PINNs across all film types (Flat, 1D, 2D). (Top) MAE values for each model indicate the average prediction errors for drug release across the three film types. Higher MAE values in the Higuchi and Peppas models for the 2D film reveal their limitations in capturing complex release patterns, while the PINN achieves low errors across all films, indicating consistent accuracy. (Bottom) RMSE values provide insight into error sensitivity to outliers. Similar to the MAE, RMSE is higher for classical models in the 2D film, highlighting challenges with structural complexity. The PINN model demonstrates robustness with low RMSE across all films, reflecting its resilience against large deviations.
• Figure 5: PINN and BPINN predictions with uncertainty bounds for simulated noisy data across flat, 1D, and 2D films. (Top left) Flat film: Both PINN (red) and BPINN (green) capture the overall release profile, with BPINN producing narrower uncertainty bands. (Top right) 1D film: PINN and BPINN predictions remain close to the data, though BPINN again shows reduced variance. (Bottom) 2D film: The PINN ensemble aligns more closely with the experimental data on average, while the BPINN provides tighter uncertainty bounds but shows a small bias. Overall, BPINNs improve robustness to noise by quantifying uncertainty, whereas PINNs may better capture mean behavior in some cases. The total time (normalized time = 1) for all release curves is 48 hours.