Topological Signatures of Diffusive Release in Porous Media

Technical Summary: Topological Signatures of Diffusive Release in Porous Media

Problem Statement
Diffusive release in porous media is governed by pore-scale geometry, including connectivity, tortuosity, and the spatial arrangement of solid obstacles. While mechanistic numerical simulations (e.g., finite element methods) are standard for studying these relationships, they are computationally expensive, particularly when screening large numbers of candidate structures for applications ranging from drug delivery to contaminant transport. There is a need for interpretable, geometry-based structural descriptors that can predict qualitative release profiles without the cost of full diffusion simulations.

Methodology
The study employs persistent homology, a tool from topological data analysis (TDA), to quantify the multiscale topology of the solid phase in porous media.

1. Simulation Setup:

   • Synthetic porous media were generated within a unit cube using spherical obstacles with varying sizes and arrangements (six structural classes: random monodisperse, random polydisperse, channel-like, layered, bottleneck-like, and clustered).
   • Samples were generated to match target porosities of 40%, 60%, and 80%.
   • Diffusive release was modeled as a mixed Dirichlet–Neumann problem on a voxelized pore space. The cumulative release curve $Q(t)$ was computed using a conforming finite element method (FEM) on a tetrahedral mesh derived from the voxel grid.

2. Topological Feature Extraction:

   • Instead of analyzing the pore space directly, the authors analyzed the solid phase ($S$) using a filtration process. The solid phase was gradually expanded (offset) by a radius $r$, creating a sequence of sets $F_r = \{x \in \Omega : \text{dist}(x, S) \leq r\}$.
   • Persistent homology was computed on this filtration to track the birth and death of topological features: connected components ($H_0$), loops ($H_1$), and voids/cavities ($H_2$).
   • Six summary statistics were extracted for each homology dimension (count, finite count, persistence sum, mean persistence, maximum persistence, and mean birth scale), resulting in 18 topological features per sample.

3. Analysis and Classification:

   • Regime Definition: Release curves were categorized into three regimes based on specific time-point metrics: "early fast" (high early release), "long tail" (high unreleased fraction at the end), and "late release" (significant increase in release during the later half of the simulation).
   • Statistical Association: Kruskal–Wallis tests were performed to determine if topological summaries differed significantly across release regimes, stratified by target porosity to isolate topological effects from porosity effects.
   • Classification: A multinomial logistic regression model was trained using only the 18 topological features to classify samples into the three release regimes.
   • Computational Timing: Wall-clock times were compared between the FEM diffusion solver and the persistent homology feature extraction pipeline.

Key Results

• Topological Signatures Independent of Porosity: Even when controlling for total porosity, samples with different release regimes exhibited statistically distinct topological signatures.

  • Early-fast release was associated with lower persistence counts and smaller persistence sums (simpler solid-phase topology).
  • Long-tail release was associated with higher persistence counts and larger persistence sums (more complex, multiscale organization of solid components, loops, and cavities).
  • The "persistence sum" emerged as the most stable and discriminative signal across all porosity levels.
  • The discriminative power of topological features decreased as porosity increased (e.g., at 80% porosity, the solid phase is sparser, making higher-dimensional features less distinct).

• Classification Performance: Using only the 18 topological features, the logistic regression model achieved test accuracies between 0.64 and 0.76 (significantly above the 0.33 chance level for a balanced three-class task). Performance was highest at 40% porosity and declined at higher porosities, consistent with the statistical association results.

• Computational Efficiency: Persistent homology feature extraction was substantially faster than FEM diffusion simulations.

  • At grid resolution $N=16$, FEM was 6.6 to 64.7 times slower than PH feature extraction.
  • At grid resolution $N=32$, FEM was 81.7 to 631.0 times slower.
  • PH extraction times remained under 0.018 seconds per sample, whereas FEM times ranged up to ~3 seconds.

Significance and Claims
The paper claims that persistent homology provides a lightweight, interpretable, and geometry-based descriptor for screening diffusive release behavior. The key contributions are:

1. Demonstrating that release behavior depends not only on the volume of pore space (porosity) but critically on the multiscale organization of the solid phase.
2. Showing that topological summaries can effectively distinguish between different release regimes (early fast, late release, long tail) without solving the diffusion equation.
3. Establishing that these topological features can be extracted at a fraction of the computational cost of traditional FEM simulations, making them suitable for early-stage prioritization of candidate porous structures before detailed simulation or experimental evaluation.

The authors position this method as a complementary tool for screening, rather than a replacement for physics-based simulations, specifically highlighting potential applications in pharmaceutical systems (e.g., sustained-release tablets) where pore architecture dictates drug release profiles.