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Multi-Trajectory Pseudotime Inference via Permutation Factorizations

Rebecca FarinaiD, Yikun WangiD, Mariano I. GabittoiD, Anamika AgrawaliD, Gonzalo E. MenaiD

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Abstract

Inferring disease progression from cross-sectional data typically relies on placing individuals along a single latent pseudotemporal axis. However, diseases often exhibit different biomarkers that evolve with different dynamics, making a single shared trajectory insufficient and often leading to biased or inconsistent orderings. We propose a probabilistic framework that maintains a shared global ordering of individuals while allowing clusters of features to follow distinct monotone trajectories along this ordering. Rather than assigning individuals to separate progressions, our approach captures heterogeneity through cluster-specific temporal responses defined with respect to a common latent sequence of individuals. Our key observation is that monotone transformations of a common progression preserve the ordering of individuals, so multiple pseudotimes factorize into a single latent permutation over individuals and cluster-specific monotonic functions, enabling flexible yet comparable representations of biomarker dynamics. We jointly infer donor ordering, biomarker clusters, and trajectories within a unified Bayesian framework, using stochastic variational inference over permutations. Across two major neuropathological datasets in Alzheimer's disease, our method recovers biologically meaningful feature clusters and improves overall alignment with established staging measures compared to single-trajectory baselines. These results show that modeling heterogeneous dynamics relative to a shared progression yields more accurate and interpretable reconstructions of disease progression from cross-sectional data.

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Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (8 Oct 2026) · CC BY 4.0

Factorizing multiple monotone pseudotimes into one shared donor permutation plus cluster-specific warps (HiP-BIND) yields Alzheimer's donor orderings agreeing better with Braak, ADNC and CERAD staging than single-trajectory B-BIND.

Existing pseudotime models for cross-sectional postmortem Alzheimer's cohorts, such as B-BIND, place all markers on one shared latent coordinate, which the authors argue forces a compromise dominated by some markers when amyloid and tau rise with differently shaped dynamics. HiP-BIND instead keeps one shared donor ordering but lets jointly inferred clusters of features follow their own monotone time warps; with a single cluster (R = 1) it closely matches B-BIND.

Key results

  1. ShownWith R = 3 groups, HiP-BIND-0 orders ROSMAP (1218 donors) and SEA-AD (84 donors) donors in closer agreement with Braak, ADNC and CERAD staging than all-marker B-BIND, without using staging labels for fitting or run selection; Thal differences are not significant.

    Median over all R = 3 runs, no selection, Braak/Thal/ADNC/CERAD: ROSMAP 0.745/0.837/0.884/0.831 vs 0.648/0.844/0.828/0.816 for B-BIND; SEA-AD 0.684/0.830/0.790/0.639 vs 0.650/0.828/0.769/0.625. Donor bootstrap (2,000 resamples): gains significant for Braak, ADNC, CERAD in both cohorts, not Thal; ROSMAP APOE carriers (279) Braak Δ = +0.184, 95% CI [0.136, 0.242]. · 4.2 Integration of different markers and…; B.2 Selection of runs; D.2 Real data

  2. SuggestedIn simulations where two feature groups share mean coefficients but differ by a nonlinear time warp, HiP-BIND recovers the donor order and both warps while B-BIND misses the second trajectory; R = 1 sharply degrades ordering, whereas extra groups leave it unchanged but split true feature groups.

    20 replicates, 100 donors, features 1–5 logistic and 6–10 power-law warp: both HiP-BIND variants beat B-BIND and PCA in Spearman with the true order in every replicate (Tables 1, 6). Going from R = 1 to R = 2 raised Spearman by 0.238 (paired 95% CI [0.220, 0.256]); larger R changed it by at most 0.003; a label-free self-consistency rule picked the true R = 2 in all 20 datasets. · 3 Simulations; Simulations; D.1 Simulations

  3. ShownFeature clusters inferred from trajectory shape separate amyloid-related features (ROSMAP amyloid and plaques; SEA-AD 6e10 and Iba1–6e10) from tau-related features (ROSMAP tau and tangles; SEA-AD AT8); resolving ROSMAP by region splits tau, grouping entorhinal cortex and hippocampus tau apart from angular gyrus tau.

    Dendrograms on 1 minus co-clustering probability over 20 HiP-BIND-0 runs. ROSMAP: amyloid, diffuse and neuritic plaques vs tau and tangles. SEA-AD: 6e10 with Iba1–6e10, NeuN joining further out, the two AT8 features apart. Region-resolved ROSMAP (EC, Hip, AG, IT): amyloid/plaques stay one cluster; EC and Hip tau/tangles group tightly, IT tau joins at larger distance, AG tau/tangles separate. · 4.3 Clustering; Region-resolved clustering (Fig. 12); Figure legends

  4. SuggestedAll-marker B-BIND orderings follow the amyloid signal: their staging correlations closely track amyloid-only B-BIND, and B-BIND fits tau as a gradual rise where HiP-BIND-0 fits a late sharp increase; in ROSMAP, tau-only B-BIND exceeds HiP-BIND on Braak, but single-class baselines score poorly on other stages.

    Table 2 (values not reproduced in the text): in both cohorts all-marker B-BIND correlations closely follow amyloid-only B-BIND; tau-only B-BIND's strongest match is Braak and amyloid-only B-BIND's is Thal, but both score poorly on the remaining stages. Figs. 3, 10, 11: SEA-AD AT8 stays flat for most of the ordering then rises late, which HiP-BIND-0 fits and B-BIND smooths; amyloid fits are similar for both models. · 4.2 Integration of different markers and…; 4.4 Recovered trajectories; Trajectories (Figs. 10, 11)

  5. SuggestedWhen only 2 of many simulated features carry ordering signal, HiP-BIND orderings stay accurate while B-BIND and PCA degrade as uninformative features are added.

    2 sigmoid signal features among 10, 20 or 40 total: with 10, B-BIND and both HiP-BIND variants reach Spearman 0.85–0.86 and PCA 0.82; as noise features increase, B-BIND drops from 0.85 to 0.34 and PCA from 0.82 to 0.56, while both HiP-BIND variants stay above 0.80 (at 40 features, run s.d. 0.14 for HiP-BIND-0 vs 0.05 for HiP-BIND-1). · E Additional simulations results

Methods, data and reproducibility

Bayesian model for cross-sectional neuropathology counts: cluster-specific pseudotimes factorize into one shared donor permutation plus Dirichlet-based increasing sequences, with soft feature-to-cluster assignments and a negative binomial likelihood. Fit in NumPyro by stochastic variational inference, with Hungarian-algorithm permutation updates (HiP-BIND-0) or a Gumbel-Sinkhorn relaxation (HiP-BIND-1).

  • SEA-AD quantitative neuropathology of middle temporal gyrus (MTG) by cortical layer: 6e10 amyloid-beta, AT8 phospho-tau, NeuN, Iba1–6e10 colocalization; Braak, Thal, CERAD, ADNC and APOE status · human (postmortem brain donors) · n = 84 donors (25 APOE carriers, 59 non-carriers)
  • ROSMAP neuropathology across eight brain regions: amyloid-beta, tau, neurofibrillary tangles, neuritic and diffuse plaques; Braak, Thal, CERAD, ADNC and APOE status · human (study participants with brain donation) · n = 1218 donors (of 4003 in release; 1645 with complete data); 279 APOE carriers
  • Simulated negative binomial counts: 10 features in two groups (logistic vs power-law warp), plus three further warp pairs · n = 20 replicates; 100 donors in the main scenario
  • Simulated noise-robustness sets: 2 signal features among 10, 20 or 40 features in total · n = 20 runs per setting
Methods & tools
HiP-BIND, NumPyro, B-BIND, PCA, Hungarian algorithm, Gumbel-Matching, Gumbel-Sinkhorn, Adam optimizer (AutoNormal guide), 6e10 immunoreactivity, AT8 immunoreactivity, NeuN, Iba1
Data & accessions
https://www.radc.rush.edu/
Assay conditions & compute
8-core Apple M2 CPU: one HiP-BIND-0 fit 31.4 min (ROSMAP) and 177 s (SEA-AD) vs 5.6 min and 26 s for B-BIND; a HiP-BIND-1 fit on ROSMAP estimated at about 14 h. Real-data run search about 130 CPU-hours for ROSMAP, run as parallel jobs on a high-performance cluster.
Reproducibility gaps
Code is said to accompany the paper (a few representative SEA-AD seeds and the simulations) but no repository URL appears in the text; ROSMAP access is controlled (approved request and Data Use Agreement); SEA-AD is under Allen Institute terms of use with no URL given.
Limitations (6)
  • Each feature's mean must be monotone along its group's pseudotime, so transient or non-monotone biomarkers may be missed; group warps are conditionally independent given the master clock, so between-group dependence beyond the shared order is not modeled. · stated by the authors
  • The inferred pseudotime is a relative ordering, not absolute disease time: only donor order is identifiable, since monotone transforms of the master clock can be absorbed by cluster warps and feature parameters. · stated by the authors
  • Without longitudinal observations the model cannot establish causal temporal progression. · stated by the authors
  • Run-to-run and bootstrap spreads are not calibrated posterior intervals: HiP-BIND-0 returns a point estimate via MAP-like permutation updates, and HiP-BIND-1's variational posterior likely understates uncertainty; calibration is left for future work. · stated by the authors
  • The B-BIND baseline is the authors' reimplementation in their own framework with a simplified coefficient prior (the original pools slopes with a full covariance across layers), so the comparison against the original B-BIND implementation is not tested. · noted in this summary
  • The simulation advantage depends on the warp pair chosen by the authors: in three further scenarios all methods reached Spearman 0.91–0.95 with differences of at most 0.018, and PCA was best in two. · noted in this summary

Who should read:Read this if you build pseudotime or disease-progression models from cross-sectional cohorts and need markers with differently shaped monotone dynamics, such as amyloid and tau, to share one donor ordering.

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