💻 computer science

PILL-CoDe: Inverse Design of Polypills via Automatic Differentiation for Prescribed Drug-Release Kinetics

PILL-CoDe is an end-to-end differentiable framework that leverages automatic differentiation to simultaneously optimize polypill geometry and excipient distribution, enabling the precise inverse design of customized drug-release kinetics through a coupled system of modified Allen-Cahn and Fickian diffusion equations.

Original authors: Rahul Kumar Padhy, Aaditya Chandrasekhar, Amir M. Mirzendehdel

Published 2026-07-17

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CC BY 4.0

Original authors: Rahul Kumar Padhy, Aaditya Chandrasekhar, Amir M. Mirzendehdel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a chef trying to bake a cake that doesn't just taste good, but also releases its sweetness at a perfectly timed rhythm: a little bit when you take the first bite, a huge burst in the middle, and a slow fade-out at the end. In the real world of medicine, this is the dream of "polypills"—single tablets packed with different medicines and helper ingredients that dissolve inside your body to release drugs exactly when they are needed. For decades, scientists have tried to control this by changing the shape of the pill (like making it a sphere or a cube) or by mixing ingredients randomly. But it's like trying to conduct an orchestra by only changing the size of the stage; you can't control which instruments play when. This paper dives into a corner of science called "inverse design," where instead of guessing and checking, you start with the desired result (the perfect release rhythm) and use powerful computers to work backward to find the perfect shape and recipe. It's a bit like asking a super-smart AI to design a custom Lego castle that, when you pull a specific string, falls apart in a pre-programmed dance rather than just crumbling into a pile.

The researchers behind this study, Rahul Kumar Padhya, Aaditya Chandrasekhar, and Amir M. Mirzendehdel, have built a digital tool called PILL-CoDe (Polypill Co-Design) to solve this puzzle. They wanted to see if they could simultaneously design the outside shape of a pill and the internal map of its ingredients to match a specific drug-release schedule. To do this, they treated the pill like a digital clay model. They used a mathematical trick called "supershapes" to define the pill's outer boundary (ensuring it stays as one solid, connected piece, not a bunch of floating crumbs) and a "neural network" (a type of AI that learns patterns) to map out exactly where different ingredients sit inside. They then simulated how the pill dissolves in water using physics equations that track how the pill's surface shrinks and how the medicine spreads out.

The team found that if you only change the pill's shape, you can only create simple release patterns, like a steady drip or a slow fade. However, when they let the AI design both the shape and the internal arrangement of ingredients, the results were magical. In their simulations, they successfully created pills that matched complex, tricky release curves, including ones that started slow, sped up dramatically, and then slowed down again. For example, in one test, the AI figured out that to get a slow start, it should hide the slow-dissolving ingredients on the outside shell, and to get a sudden burst of medicine later, it should pack the fast-dissolving ingredients deep in the center. When the outer shell wore away, the fast ingredients were suddenly exposed, creating the desired spike in the release rate. The computer simulations showed that this multi-material approach could match these complex targets with incredible accuracy (getting within 1% of the goal), whereas trying to do it with just one type of material and changing only the shape failed miserably.

The paper also explored how the starting point of the design matters. They tried starting with a "spiky" shape, a "circle," and a "sunflower" shape. Surprisingly, the AI found different internal maps for each starting shape, yet all of them ended up producing the exact same perfect release curve. This suggests there isn't just one "right" answer, but many different ways to build a pill that works the same way, giving engineers flexibility to choose designs that are easier to print or easier for patients to swallow. They even tested a scenario where some of the ingredients were "aged" or degraded (like medicine stored for a long time in space), and the AI successfully adjusted the recipe to compensate, ensuring the pill still worked correctly.

However, it is important to remember that these results are currently just simulations. The authors used a computer to solve complex physics equations involving how the pill dissolves and how the medicine diffuses, but they haven't physically printed and tested these specific pills in a lab yet. The paper explicitly rules out the idea that changing the shape alone is enough for complex needs; they showed that without mixing different materials inside, you simply cannot achieve those fancy, non-linear release patterns. While the math suggests this method works perfectly, the authors note that real-world factors like how the pill interacts with stomach acid, how the body absorbs the drug, and the physical limits of 3D printers are things that still need to be tested. But for now, PILL-CoDe offers a promising new way to think about medicine: not just as a static object, but as a dynamic, programmable machine that can be designed from the inside out to heal us exactly when we need it.

Technical Summary: PILL-CoDe – Inverse Design of Polypills via Automatic Differentiation

Problem Statement

Polypills—single oral dosage forms combining multiple active pharmaceutical ingredients (APIs) and excipients—offer the potential for personalized, fixed-dose combination therapies with coordinated multi-phase release. While additive manufacturing (AM) enables the physical realization of complex multi-material geometries, current formulation workflows rely on ad hoc parameter sweeps. These methods are ill-suited for systematically exploring the high-dimensional design space of shapes, compositions, and release behaviors.

Existing inverse-design approaches suffer from three primary limitations:

Decoupled Optimization: They typically optimize geometry and material composition separately, failing to co-optimize the pill's external shape with the spatial distribution of multiple active ingredients and excipients.
Simplified Physics: Many models treat dissolution as a geometric front-propagation problem (governed by Eikonal equations) rather than a coupled dissolution-diffusion transport process. This neglects critical phenomena such as local concentration buildup, saturation effects, and diffusion-limited transport, which are central to classical dissolution theory (e.g., Noyes-Whitney and Nernst-Brunner models).
Computational Barriers: Embedding complex, coupled physics into optimization loops often requires laborious manual derivation of adjoint sensitivities, which becomes increasingly cumbersome as physical fidelity increases.

Methodology

The authors present PILL-CoDe, a framework for the simultaneous co-design of polypill geometry and excipient distribution to match prescribed drug-release kinetics. The framework is implemented in JAX, ensuring the entire pipeline is end-to-end differentiable.

1. Hybrid Design Representation

To balance manufacturability (topological connectivity) with design freedom (material heterogeneity), the framework employs a hybrid parameterization:

Geometry (Supershapes): The global pill geometry is parameterized using supershapes (Gielis curves). This ensures the resulting dosage form is a single, connected, monolithic domain. The geometry is defined by seven parameters: translation ( $c_x, c_y$ ), rotation ( $\theta$ ), semi-axis scaling ( $a, b$ ), squareness ( $n$ ), and rotational symmetry ( $m$ ).
Material Distribution (Neural Implicit): The internal distribution of $S$ distinct excipients is represented by a coordinate-based neural network. The network takes spatial coordinates as input and outputs the volume fraction of each excipient. A Softmax activation enforces the partition of unity constraint ( $\sum \gamma_s = 1$ ). This approach decouples the design resolution from the simulation mesh and allows for high-resolution, continuous material fields.

2. Physics-Informed Forward Model

The dissolution process is governed by a coupled system of partial differential equations (PDEs):

Interface Evolution: A modified Allen-Cahn phase-field equation models the moving solid-solvent interface. It includes a dissolution forcing term derived from Nernst-Brunner kinetics, which couples the phase evolution to the local solute concentration.
Mass Transport: Fickian diffusion governs the transport of the dissolved species in the solvent. The source term for diffusion is coupled to the phase-field interface, ensuring mass conservation.
Boundary Conditions: The system assumes an idealized in vitro sink condition (zero concentration at boundaries) and zero-flux conditions for the phase field.

3. Optimization Framework

The inverse design problem is formulated as minimizing the Mean Squared Error (MSE) between the simulated instantaneous mass release rate and a prescribed target profile.

Constraints: The optimization enforces:
- Grayness Constraint: A continuation strategy drives the continuous material fractions toward binary states (0 or 1) to ensure physical realizability.
- Volume Constraints: Minimum volume fractions for specific excipients are enforced using a smooth minimum formulation.
- Box Constraints: Geometric parameters are bounded to ensure physical meaningfulness.
Sensitivity Analysis: The framework utilizes Automatic Differentiation (AD) via JAX to compute exact sensitivities. To manage the computational cost of transient, nonlinear solvers, the authors employ:
- The Implicit Function Theorem (IFT) to bypass backpropagation through Newton-Raphson iterations.
- Checkpointing to mitigate memory bottlenecks associated with storing state history for long transient simulations.

Key Contributions

The paper identifies four specific contributions:

Hybrid Parameterization: A novel design space combining supershapes for global geometry and coordinate-based neural networks for internal multi-material distribution.
Differentiable Dissolution-Diffusion Simulator: A physics-informed model coupling a modified Allen-Cahn phase-field equation with Fickian diffusion, capturing local concentration buildup and saturation effects absent in front-propagation models.
End-to-End Inverse Design Framework: A formulation where governing equations are embedded directly in the optimization loop, with sensitivities computed automatically via the discretized solver, avoiding manual adjoint derivation.
Expanded Reachable Set: Demonstration that co-designing geometry and excipient distribution significantly expands the set of achievable release profiles compared to geometry-only optimization, enabling the matching of complex, non-monotonic kinetics.

Results

The framework was validated through numerical experiments on a 2D domain using a 75×75 mesh and a 2-layer neural network.

Validation of Hypothesis:
- Monotonic Targets: Single-material optimization (geometry only) successfully matched simple, monotonic release profiles.
- Non-Monotonic Targets: Single-material designs failed to match complex profiles requiring an initial slow release, rapid acceleration, and sharp decay (Normalized MSE: $3.01 \times 10^{-2}$ ). In contrast, the multi-material co-design successfully matched these profiles (Normalized MSE: $9.52 \times 10^{-5}$ ).
- Mechanism: The optimizer placed slow-dissolving excipients at the periphery to suppress early release and fast-dissolving excipients in the interior to create a burst upon shell erosion.
Initialization Robustness: The framework converged to equivalent functional performance from diverse initial geometries (e.g., "spike," "circle," "sunflower"), demonstrating robustness against local minima and allowing designers to select shapes based on manufacturability or ingestibility.
Degradation Case Study: The framework was applied to a scenario involving degraded feedstock (simulating spaceflight conditions). By varying minimum volume constraints for fresh, moderately aged, and heavily aged excipients, the optimizer successfully adapted the material distribution to match target release profiles, validating its utility for resource-constrained environments.

Significance and Claims

The paper claims that PILL-CoDe addresses a critical gap in the systematic design of controlled-release polypills. By moving beyond simplified front-propagation models to a coupled dissolution-diffusion framework, the method captures the physical mechanisms (concentration gradients, saturation) that govern release in multi-material systems.

The authors assert that their approach enables the systematic co-design of shape and composition under manufacturability constraints, a capability previously hindered by the complexity of deriving sensitivities for coupled physics. The results suggest that multi-material inverse design is essential for tailoring release kinetics to complex, patient-specific targets that cannot be achieved by geometry alone.

The authors remain modest regarding the scope of their claims, noting that:

The current model is restricted to dissolution-dominated regimes under idealized sink conditions.
The reachable set of release profiles is bounded by the interplay of geometry, material distribution, and physical constraints (e.g., non-negative release rates).
Practical implementation requires future work on experimentally validated material datasets, 3D extensions, and the incorporation of additive manufacturing constraints (e.g., minimum feature size) and in vivo physiological factors.