TI$^2$PS: A Topology-Informed Inverse Design Framework for Stochastic Multicellular Pattern Formation

Kenji Komiya; Andrew Kailiang Jin; Ryo Nishikimi; Kunio Kashino

cs.LGAug 28, 2026

TI $^2$ PS: A Topology-Informed Inverse Design Framework for Stochastic Multicellular Pattern Formation

Authors: Kenji Komiya, Andrew Kailiang Jin, Ryo Nishikimi, Kunio Kashino

Organizations: NTT, Inc., Kanagawa 2430124 Japan · Georgia Institute of Technology, Atlanta, GA 30332 USA

Abstract

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.

Figures & tables

Fig. 1: Proposed agent-based model (ABM) optimization workflow. (a) A training and testing dataset of the desired ABM is generated through hypercube sampling. (b) Cellular ABM outputs can be generically quantified using topological data analysis, resulting in a series of Betti vectors that represent the topological features of the ABM output. (c) Surrogate model training and testing using quantified model outputs (Betti vectors) and known input parameter values. (d) Target patterns can be queried into the trained model to elucidate parameter combinations that may yield similar patterns in simulation.

Fig. 2: An example of ABM simulation results. All parameters for the ABM simulation were taken from [ 27 ] .

Fig. 3: Visual comparisons of simulated patterns using parameters estimated by each method. The leftmost column shows seven representative target patterns from the test set, including stripes (rows a–e) and dots (rows f and g). These targets exhibit variations in cell distribution, such as black-dominant (row a), yellow-dominant (rows d and e), and balanced (rows b and c) patterns. Columns 2–7 show simulation results based on the parameters estimated by the proposed and conventional methods. The second and third columns represent the proposed TI 2 PS GLM and TI 2 PS MLP, while the remaining columns show results from the conventional methods. All simulated patterns, except for the target pattern, were generated using the corresponding models trained with random seed 0.

Method	MSE ( $\downarrow$ )	Pearson $r$ ( $\uparrow$ )	$\sigma_{\mathrm{ratio}}$ (%) ( $\uparrow$ )
TI 2 PS GLM (Ours)	$0.2476$ $\pm 0.0021$	$0.4933$ $\pm 0.0068$	$53.80717$ $\pm 0.4956$
TI 2 PS MLP (Ours)	$\mathbf{0.1911}$ $\mathbf{\pm 0.0016}$	$\mathbf{0.6360}$ $\mathbf{\pm 0.0038}$	$\mathbf{68.5540}$ $\mathbf{\pm 1.5241}$
Pooling GLM	$0.3533$ $\pm 0.0003$	$-0.0039$ $\pm 0.0015$	$18.2502$ $\pm 0.1324$
Pooling MLP	$0.3677$ $\pm 0.0009$	$-0.0008$ $\pm 0.0005$	$27.7520$ $\pm 0.4176$
PointNet	$0.351$ $\pm 0.0006$	$-0.0054$ $\pm 0.0006$	$16.2239$ $\pm 0.5604$
PointNet++	$0.4203$ $\pm 0.0048$	$0.0019$ $\pm 0.0018$	$49.1806$ $\pm 1.3653$

TABLE I: Performance comparison in terms of MSE, Pearson correlation (

r

), and standard deviation ratio (

\sigma_{\mathrm{ratio}}

). Values are reported as the mean

\pm

standard deviation over five independent runs with different random seeds.

Fig. 4: Sensitivity analysis of the training data amount for the proposed TI 2 PS MLP and PointNet++ methods: (a) MSE, (b) Pearson correlation

r

, and (c) standard deviation ratio

\sigma_{\mathrm{ratio}}

. The proposed TI 2 PS MLP method shows superior performance and stability across all metrics, even with limited training data. Markers indicate the mean values over five independent runs with different random seeds, and the shaded regions indicate the corresponding standard deviation ranges.

Fig. 5: Validation result with manually crafted mutant data: (a-1) target distorted stripe pattern of dali/+ mutant and (a-2) simulated pattern using the parameters estimated from the target; (b-1) target polka-dot-like pattern of leopard mutant and (b-2) simulated pattern using the parameters estimated from the target.

Parameter	Wild type (stripe)	dali/+	leopard
$R^{BB}$	250.00	941.64	858.84
$r^{BB}$	20.00	8.89	0.38
$R^{YY}$	200.00	938.39	901.65
$r^{YY}$	11.00	8.88	77.44
$R^{BY}$	450.00	214.23	813.95
$r^{BY}$	20.00	52.47	94.87

TABLE II: Estimated parameters from wild type and mutant patterns.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

$\epsilon_{i}-\epsilon_{i-1}$	MSE ( $\downarrow$ )	Pearson $r$ ( $\uparrow$ )	$\sigma_{\mathrm{ratio}}$ (%) $\uparrow$ )
0.01	$0.2905$ $\pm 0.0006$	$0.3627$ $\pm 0.0030$	$40.9377$ $\pm 0.6309$
0.05	$0.2248$ $\pm 0.0023$	$0.5532$ $\pm 0.0062$	$60.7877$ $\pm 0.9723$
0.10	$\mathbf{0.1911}$ $\mathbf{\pm 0.0016}$	$\mathbf{0.6360}$ $\mathbf{\pm 0.0038}$	$\mathbf{68.5540}$ $\mathbf{\pm 1.5241}$

Appendix

TABLE III: Sensitivity analysis of Betti-vector filtration discretization. The discretization step is defined as

\epsilon_{i}-\epsilon_{i-1}

. Values are reported as the mean

\pm

standard deviation over five independent runs with different random seeds.

Method	MSE ( $\downarrow$ )	Pearson $r$ ( $\uparrow$ )	$\sigma_{\mathrm{ratio}}$ (%) ( $\uparrow$ )
TI 2 PS GLM (Averaged)	$0.2476$ $\pm 0.0021$	$0.4933$ $\pm 0.0068$	$53.8072$ $\pm 0.4956$
TI 2 PS GLM (Non-averaged)	$0.2408$ $\pm 0.0007$	$0.5135$ $\pm 0.0051$	$54.4068$ $\pm 0.4590$
TI 2 PS MLP (Averaged)	$0.1911$ $\pm 0.0016$	$0.6360$ $\pm 0.0038$	$68.5540$ $\pm 1.5241$
TI 2 PS MLP (Non-averaged)	$0.1940$ $\pm 0.0031$	$0.6169$ $\pm 0.0039$	$77.6200$ $\pm 0.8681$

Appendix

TABLE IV: Impact of averaging Betti vectors for TI 2 PS GLM and TI 2 PS MLP. Values are reported as the mean

\pm

standard deviation over five independent runs with different random seeds.

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