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.
Controlled release plays a pivotal role in modern medicine and pharmacology, offering the potential to target therapeutic agents with precision, thereby optimizing efficacy and minimizing side effects. The controlled release of drugs is essential in numerous clinical contexts, from cancer therapy [1] to the management of chronic diseases [2]. Effective drug delivery systems ensure that drugs reach their intended targets at the correct dosage and over the appropriate time frame, a critical factor in improving patient outcomes. The design and simulation of such systems rely heavily on modeling drug transport mechanisms within various substrates, including polymers and tissues [3].
The mathematical modeling of drug release is grounded in several classical models, with Fick’s Law of Diffusion, Higuchi’s Model, and the Peppas-Korsmeyer Model being some of the foundational approaches. Fick’s Law describes the diffusion process based on concentration gradients and is widely used to model drug transport through porous matrices [4]. The Higuchi model, initially developed for planar systems, expands upon Fickian diffusion to describe the release of drugs from thin films, taking into account the square-root time dependency [5]. Peppas’s Model, on the other hand, generalizes drug release behavior, including both Fickian and non-Fickian diffusion processes, allowing for more versatile characterization of drug release kinetics in various geometries and materials [6]. These models have served as the backbone of drug release modeling for decades, providing analytical solutions for specific release system geometries.
However, real-world drug release systems are often more complex, necessitating the use of modeling assumptions and numerical methods to solve partial differential equations (PDEs) where analytical solutions are not feasible [7]. Traditional numerical methods, such as finite element and finite difference methods, have been applied to solve these models under different boundary and initial conditions [8]. The limitations of these classical approaches become more clear in complex material systems, such as hydrogels, where factors such as polymer concentration and structural interactions influ-ence drug release [9]. Recently, neural networks have demonstrated success in addressing broader mechanistic problems [10,11], as well as in applications related to targeted drug delivery [12] [13]. In particular, Physics-Informed Neural Networks (PINNs), have emerged as powerful tools for solving initial and boundary value mechanistic problems [14] [15,16]. PINNs leverage deep learning to integrate physical laws, described by differential equations, into the training process, allowing neural networks to approximate solutions to complex PDEs [17]. In addition to PINNs, Bayesian Physics-Informed Neural Networks (BPINNs) provide a framework for handling noisy data by incorporating uncertainty quantification into the predictions, which is crucial when dealing with real-world, imperfect data [18]. This allows BPINNs to simultaneously solve forward and inverse problems with greater robustness. Our main contribution is to propose a method beyond classical drug release models that integrates a small amount of real-life/experimental data with physical laws to inform neural networks that can then make accurate drug release predictions. This hybrid approach (i) achieves higher predictive accuracy than traditional methods and (ii) requires significantly fewer data points, enabling reliable drug release modeling even under limited experimental conditions.
To address the inverse problem in drug delivery, which is often ill-posed and challenging to solve using conventional methods, we leverage PINNs. Inverse problems typically involve determining unknown parameters or functions from observed data. These inverse problems have been applied to mechanics [19], but in drug delivery, this data can be incomplete, noisy, or poorly defined, making traditional solutions unstable or unreliable. PINNs offer a solution by enabling a “one-shot” approach, where the neural network learns both the solution to the differential equation and the parameters of interest simultaneously. However, in scenarios where experimental data has significant noise, the deterministic nature of PINNs can lead to large variances and less accurate results. This is where BPINNs can help by incorporating Bayesian inference to manage uncertainty and improve model robustness in the presence of noise. Unlike traditional numerical solvers, which struggle with noisy data, BPINNs enable uncertainty quantification, providing a probabilistic framework that can yield reliable solutions even with imperfect data. The Bayesian approach, though computationally intensive due to methods like Hamiltonian Monte Carlo (HMC) sampling, offers significant advantages in handling noisy, real-world experimental data, especially in higher-dimensional problems [18].
This study leverages the experime
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