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TI$^2$PS: A Topology-Informed Inverse Design Framework for Stochastic Multicellular Pattern Formation
Kenji Komiya, Andrew Kailiang Jin, Ryo Nishikimi, Kunio Kashino
TL;DR
Multicellular ABMs need better ways to estimate cell-level parameters and compare stochastic patterns whose cell numbers and arrangements vary. TI2PS addresses this by combining Betti vectors from topological data analysis with inverse surrogate modeling, and in zebrafish pigment-pattern simulations it outperformed PointNet++ even with one-tenth as much training data.
Problem
ABMs make it difficult to estimate agent-specific parameters and quantitatively compare multicellular patterns when stochastic proliferation and death change cell number and arrangement.
Method
TI2PS uses Betti vectors derived from TDA to encode global cellular structure and an inverse surrogate model to infer ABM parameters from observed patterns.
Results
Using 10% of the training data, TI2PS outperformed PointNet++ using 100% across all evaluation metrics.
Takeaways & Limitations
Betti vectors provide a topological feature representation for estimating ABM parameters from stochastic multicellular patterns.
Takeaways & Limitations
Mutant-pattern reconstruction failures could reflect target fidelity, parameter-range coverage, missing biological mechanisms, inverse-model limitations, or other factors, so the experiment is exploratory.
Abstract
from arXiv · showhide
This study proposes a novel framework to estimate parameters for reproducing target multicellular patterns using an agent-based model (ABM). Two major challenges in multicellular ABMs are estimating cell-level parameters (agent-specific variables) and quantitatively evaluating the topological characteristics of multicellular arrangements under stochastic cell proliferation and death. To address these challenges, we integrate two approaches: Betti vectors and inverse surrogate modeling. The Betti vectors obtained through topological data analysis can consistently represent features of a wide range of multicellular spatial configurations. The inverse surrogate modeling enables direct inference of the corresponding ABM parameters from the target patterns. We validated the proposed framework using zebrafish pigment pattern formation, a representative model of pattern formation driven by multicellular interactions. The results demonstrate that our framework successfully estimates ABM parameters and outperforms conventional methods such as PointNet++. Notably, the proposed method, which used only 10% of the training data, outperformed PointNet++, which used 100% of the data, across all evaluation metrics.
I. INTRODUCTION
Multicellular pattern formation is important for tissue development, repair, disease understanding, and engineered biological systems, but ABMs struggle with agent-specific parameter estimation and stochastic cell-number variation. TI2PS combines inverse surrogate modeling with TDA to infer parameters from global pattern structure and is evaluated on zebrafish pigment formation.
- Significance of elucidating mechanisms of multicellular behavior: Multicellular pattern formation supports tissue development, repair, regeneration, disease research, and engineered tissues or organs.Cells self-organize through cell-cell interactions and environmental cues; abnormal patterns can accompany cancer and genetic disorders.
- Vitalization of computer simulation of cellular behavior: ABMs represent individual cells as information-processing agents governed by rules dependent on the environment and other cells.They provide a computational approach for studying multicellular behavior across scales.
- Current challenges for computational modeling of cellular behavior: Agent-specific parameter estimation is difficult because grid searches become impractical in high-dimensional spaces when each cell-level simulation is computationally intensive.This challenge limits efficient exploration of ABM parameter spaces.
- Current challenges for computational modeling of cellular behavior: Stochastic cell proliferation and death vary model complexity and agent count, increasing the difficulty and instability of estimating cell-level parameters.These estimation and stochasticity problems are interdependent.
- Our key strategy: TI2PS combines inverse surrogate modeling, which estimates simulator parameters from patterns, with TDA, which compares global structure despite differing feature number, position, or scale.The framework explicitly integrates inverse parameterization and topology in an ABM pipeline.
- Our key strategy: The framework was evaluated on zebrafish pigment pattern formation using interacting cell types whose patterns arise from migration, division, and death.The inverse surrogate models included a generalized linear model and a multilayer perceptron.
III. METHOD
The method maps observed multicellular patterns to simulator parameters through Betti-vector features derived from persistent homology. It constructs a Vietoris-Rips filtration, tracks connected components and loops across scales, forms Betti curves, and concatenates them into a machine-learning input.
- A. Problem Specification: The inverse surrogate model takes observed cell positions for each cell type as input and outputs the biological simulator’s parameters.The parameters are indexed by pairs of same or different cell types to represent their relationships.
- B. Betti Vector: A Vietoris-Rips filtration constructs simplicial complexes from balls of radius ϵ/2 centered at cell positions, adding simplices when ball intersections occur.Points, line segments, and filled triangles correspond to 0-, 1-, and 2-simplices.
- B. Betti Vector: Persistent homology tracks the birth and death of connected components and loops as the filtration value ϵ increases.Components merge, while loops can emerge and disappear when filled by triangles.
- B. Betti Vector: For each topological degree, birth-death pairs record when features appear and disappear, providing the basis for Betti curves.The feature counts are derived from homology across filtration values.
- B. Betti Vector: Betti curves return the number of connected components and loops alive at each filtration value using an indicator function.The indicator equals 1 when the specified condition holds and 0 otherwise.
- B. Betti Vector: The full Betti vector concatenates Betti-curve vectors across topological degrees and cell types for downstream statistical or machine-learning analysis.In this study, it is used as the inverse surrogate model’s input feature, with 1000 filtration values starting at 0 and spaced by 0.1 µm.
C. Proposed TDA-based Inverse Surrogate Methods
The proposed inverse surrogate methods infer simulator parameters from Betti vectors representing observed biological patterns. A GLM provides a direct mapping, while an MLP captures more complex dependencies.
- The methods estimate simulator parameters Θ from the full Betti vector v of an observed biological pattern.
- The GLM maps v to Θ using a parameterized transformation with an element-wise nonlinear function.
- The MLP extends this approach with L hidden layers to capture more complex dependencies between Θ and v.
- Both the GLM and MLP use the hyperbolic tangent as their nonlinear activation function.
D. Conventional Methods
The conventional methods estimate simulator parameters from permutation-insensitive summaries of cell-position point clouds. They include hand-crafted pooling features and PointNet architectures, with PointNet++ adding hierarchical local-geometry processing.
- Because cell counts vary and point order is noninformative, conventional methods must handle variable-sized, unordered point clouds.
- 1) Statistical pooling-based method:: Statistical pooling summarizes each cell type’s coordinates using maximum, minimum, and average values for every dimension.
- 1) Statistical pooling-based method:: PoolingGLM and PoolingMLP map the concatenated fixed-size pooling features s to simulator parameters Θ.
- 2) PointNet-based method:: PointNet maps cell positions into latent features with an MLP and applies max-pooling to achieve permutation invariance before estimating Θ.
- 2) PointNet-based method:: PointNet++ hierarchically captures local geometric structures through set-abstraction layers using farthest-point sampling and neighborhood aggregation.
E. Optimization
The inverse surrogate models are trained on paired cell-position features and ground-truth simulator parameters by minimizing squared prediction errors. Adam is used for optimization.
- The training dataset contains N samples pairing cell positions X_n with ground-truth simulator parameters Θ̂_n ∈ R^DΘ.
- Each model converts X_n into an input feature vector a_n and optimizes its parameters by minimizing the sum of squared errors.
- Adam is used as the optimizer for all evaluated methods.
IV. EXPERIMENTS
Experiments use a stochastic zebrafish pigment-pattern ABM with two interacting cell types, growth, movement, birth, and death. Patterns generated from sampled parameters serve as targets for inverse estimation.
- The ABM represents black melanophores and yellow xanthophores, whose interactions produce zebrafish stripe patterns.
- The simulation initializes cell positions on horizontal lines and stretches the domain and rearranges cells as development progresses.
- Between timesteps, neighborhood-dependent probabilistic birth and death events respectively increase and decrease the cell-position sets.
- Cell interactions are modeled with Morse potentials whose strength and length parameters define repulsion and attraction between cell types.
2) Manually Crafted Zebrafish Stripe:
Manually crafted zebrafish mutant patterns were represented by approximately 6,000 manually placed cells, with cell types specified for each placement.
- Approximately 6,000 cells were manually placed within an ABM-sized domain for the dali/+ and leopard mutant patterns.The dali/+ pattern contained 5,451 cells, while leopard contained 6,204 cells.
B. Parameter Settings
The ABM simulations used predefined domain, cell-position, proliferation, death, and training hyperparameters to reproduce zebrafish patterns.
- The simulation domain started at w0 = 2 mm and h0 = 1 mm and expanded by k = 130 µm per day.
- Training normalized inputs to mean 0 and standard deviation 1, scaled targets to [−1, 1], and minimized MSE with Adam and early stopping.Early stopping used validation-loss patience of 10 epochs.
- All models were trained with batch size 512, while MLP-based models used three hidden layers with 1024 units each.
- Experiments were repeated five times with random seeds 0, 1, 2, 3, and 4 to assess stochastic variability.
C. Evaluation Metrics
The inverse surrogate model was evaluated using MSE, Pearson correlation, and the ratio of estimated-to-true parameter standard deviations.
- Three metrics evaluated inverse parameter estimation: mean squared error, Pearson correlation coefficient r, and standard deviation ratio σratio.
- MSE compares estimated and true parameters, with the metric computed after normalization to the training-set range [−1, 1].
- Pearson correlation r measures association between estimated and true parameters and is averaged across parameters and cell-type pairs.
- σratio compares the standard deviation of estimated parameters with that of true parameters and is averaged across parameters and cell-type pairs.
D. Results
TI2PS reproduced major zebrafish pattern features and achieved stronger parameter-estimation performance than conventional methods, including with substantially less training data.
- 1) Validation with Simulated Data: TI2PS GLM and MLP mostly reproduced large-scale stripes and polka dots, although local and occasional global discrepancies remained.
- 1) Validation with Simulated Data: Conventional methods often failed to reproduce even the target patterns’ large-scale features in representative examples.
- Sensitivity to Training Data Size: With only 10% of the training data, TI2PS MLP outperformed full-data PointNet++ across all evaluation metrics.
- Sensitivity to Training Data Size: TI2PS MLP’s correlation r increased with training-data size, whereas PointNet++ remained near zero with little change.
- 2) Manually Crafted Zebrafish Stripe: TI2PS MLP reproduced several crafted mutant features, including dominant xanthophores, incomplete dali/+ stripes, and small leopard melanophore clusters.
- 2) Manually Crafted Zebrafish Stripe: Differences remained in mutant simulations, including larger inter-type gaps and smaller leopard melanophore clusters than in crafted patterns.
3) Validation with Manually Crafted Mutant Patterns:
TI2PS estimates ABM parameters from spatial patterns using TDA-derived Betti vectors and inverse surrogate modeling. Validation shows strong performance on standard patterns, while manually crafted mutant reconstructions remained incomplete and diagnostically ambiguous.
- Framework: TI2PS estimates ABM parameters from spatial cell patterns through a TDA-derived Betti-vector feature.The framework uses Betti vectors to quantify global pattern structure for inverse surrogate modeling.
- Standard-pattern validation: TI2PS MLP outperforms all baseline methods across MSE, Pearson correlation r, and output variance ratio σratio.The reported comparison includes both quantitative metrics and visual reproduction of stripes and dots.
- Standard-pattern validation: Pooling-based models produce near-zero correlation and fail to recognize unique target features because pooled coordinates do not capture geometric structure.Their predictions tend toward average parameters, yielding polka-dot-like outputs for many targets.
- Standard-pattern validation: PointNet and PointNet++ retain near-zero Pearson correlation r despite improving MSE and σratio, indicating weak tracking of actual parameter changes.The discussion attributes this to difficulty extracting distinctive features from spatially uniform or periodic cell patterns.
- Data-efficiency validation: Using 10% of the training data, TI2PS MLP outperformed PointNet++ trained on 100% across all evaluation metrics.The sensitivity analysis reports this result across MSE, Pearson correlation, and standard deviation ratio.
- Validation with manually crafted mutant patterns: Mutant-pattern reconstructions captured several features but did not fully reproduce the manually crafted dali/+ and leopard patterns.The study cannot determine whether discrepancies arise from target fidelity, parameter-range limits, missing biology, or inverse-model limitations.
APPENDIX I SENSITIVITY ANALYSIS OF BETTI-VECTOR FILTRATION
The sensitivity analysis examines filtration-step discretization and stochastic-realization averaging in TI2PS Betti-vector construction. Larger filtration steps improved tested metrics, while removing averaging largely preserved the proposed methods’ performance trend.
- Filtration sensitivity: Increasing the filtration step size improved all evaluation metrics in the tested settings with NE = 1000 filtration values.The comparison varied ϵi − ϵi−1 while keeping the number of filtration values fixed.
- Averaging sensitivity: Table IV reports means ± standard deviations over five independent runs with different random seeds.This reporting design evaluates variability associated with the training procedure.
- Averaging sensitivity: Removing Betti-vector averaging changed MSE and Pearson correlation only marginally for TI2PS GLM and TI2PS MLP.The averaged setting used five stochastic ABM realizations per parameter set, whereas the non-averaged setting treated each realization as an individual sample.
- Averaging sensitivity: Although σratio differed more for TI2PS MLP without averaging, the overall performance trend was preserved.The results indicate that the proposed methods’ advantage is not solely attributable to averaging stochastic realizations.