Latent Disease Pseudotime: Modeling Progression

• Latent disease pseudotime is a model that re-aligns patients along a hidden progression axis, separating disease stage from clock time using various mathematical formulations.
• It encompasses methods such as sigmoidal latent states, patient-specific time shifts, deep state-space models, and graph-based trajectories to robustly capture disease evolution.
• Its framework offers interpretable staging and mechanistic readouts, though its accuracy and clinical relevance are sensitive to model assumptions, data quality, and anchor selection.

Latent disease pseudotime is an unobserved disease-stage coordinate that realigns patients, subjects, or subject-level representations onto a progression axis that is more closely related to pathology than to chronological age, study entry, or irregular visit timing. Across the recent literature, it appears in several mathematically distinct forms: a latent disease state evolving in a probabilistic state-space model, a latent disease time or latent disease age that shifts observed trajectories, a non-decreasing hidden stage sequence, a graph-level trajectory over pathway graphs, a geodesic distance on a branching clinical manifold, or a scalar stage variable used to parameterize stage-varying causal effects (Nakamura et al., 2023, Lespinasse et al., 2023, Li et al., 2017, Shi et al., 10 Feb 2025, Golovenkin et al., 2020, Le, 14 Jun 2026). What unifies these formulations is the attempt to recover disease progression from noisy, incomplete, heterogeneous observations while separating disease stage from raw clock time.

1. Conceptual definition and scope

Latent disease pseudotime treats progression as an internal disease clock rather than as elapsed chronological time. In Alzheimer’s disease, this distinction is explicit: one line of work models a latent variable $Z$ that “orders patients along a data-driven disease trajectory independent of chronological age,” while another defines latent disease time as an individual-specific realignment relative to a true but unobserved clinical onset $T_i^*$, so that disease time is zero at dementia onset and negative before diagnosis (Glazman et al., 6 Nov 2025, Lespinasse et al., 2023). In related Bayesian mixed-effects work, the same idea appears as a subject-specific latent time shift $\delta_i$ added to observed follow-up time, producing a common long-term disease timeline from short-term multicohort data (Li et al., 2017).

The literature does not restrict latent disease pseudotime to a single scalar construction. Some models use a continuous one-dimensional coordinate, such as the sigmoidal latent disease state $d_{ij}^k \in [0,1]$ for each clinical target in progressive disease forecasting (Zhu et al., 2018). Others use multivariate latent trajectories $z_{1:T}$ or $Z_t$, where the path through latent space, rather than a single number, is the progression representation (Trottet et al., 2024, Nakamura et al., 2023). Still others use discrete latent stages with a monotonicity constraint $s_i \le s_{i+1}$, so that the stage index itself acts as an ordered latent coordinate (Zaballa et al., 2023). In cross-sectional settings, pseudotime may be a geodesic distance from a root state on a principal tree or a graph-level trajectory over subject pathway graphs (Golovenkin et al., 2020, Shi et al., 10 Feb 2025).

Several papers explicitly distinguish these disease models from classical single-cell pseudotime. The distinction is not that disease pseudotime abandons ordering, but that it transfers ordering to patients, subject graphs, pathology embeddings, or longitudinal disease states. In this sense, the field encompasses both direct latent-time models and pseudotime-like embeddings whose trajectories are interpreted as disease progression (Shi et al., 10 Feb 2025, Nakamura et al., 2023, Trottet et al., 2024).

2. Formal model classes

The formalizations of latent disease pseudotime differ in how they represent progression, but they share the structure of a hidden state or hidden coordinate linked to observed data.

Model family	Latent progression object	Representative relation
Probabilistic progression model	Sigmoidal latent disease state	$$d_{ij}^k = \frac{1}{1+\exp\big(-(t_{ij}-p_i^k)s_i\big)}$$
Diagnosis-anchored latent time	Shifted disease time	$s_i(t)=t-T_i^*$
Latent disease age joint model	Shifted and rescaled latent time	$s_i(t)=\exp(\xi_i)(t-\tau_i)$
Deep state-space model	Time-varying latent state	$T_i^*$0
Graph-level pseudotime	Subject-level graph trajectory	$T_i^*$1
Stage-aware Bayesian network	Scalar latent disease stage	$T_i^*$2

In probabilistic disease progression models, the latent coordinate is often explicitly monotone. A widely cited formulation defines, for subject $T_i^*$3, visit $T_i^*$4, and target $T_i^*$5, a latent disease state

$T_i^*$6

where $T_i^*$7 is a target-specific inflection point and $T_i^*$8 is a subject-specific slope shared across targets. Observed scores are then noisy linear functions of this latent axis,

$T_i^*$9

so the latent sigmoid acts as a continuous disease pseudotime coordinate that aligns different patients independently of their observation schedule (Zhu et al., 2018).

A second family represents pseudotime as a patient-specific time shift. In diagnosis-anchored dementia modeling, latent disease time is

$\delta_i$0

where $\delta_i$1 is the true but unobserved time of clinical dementia onset. In latent disease age models for ALS, the observed clock is transformed by both a shift and an individual rate factor,

$\delta_i$2

allowing subjects to differ in both position and speed along a common disease axis (Lespinasse et al., 2023, Ortholand et al., 2024).

A third family uses latent states evolving through explicit temporal dynamics. In the deep state-space analysis framework for EHR time series, each patient has a latent state $\delta_i$3 at every time step, with transition model $\delta_i$4 and emission model $\delta_i$5. The model is trained with an ELBO and standard normal initial prior $\delta_i$6, so the latent trajectory is learned from observed EHR sequences alone (Nakamura et al., 2023). Related deep generative models for systemic sclerosis use a temporal latent process $\delta_i$7 with a factorized Gaussian prior conditioned on observation times, medications, and demographics, and augment the ELBO with medically guided supervision (Trottet et al., 2024).

A fourth family defines pseudotime over graphs or manifolds inferred from cross-sectional data. Graph-level Pseudotime Analysis embeds subject pathway graphs, constructs a K-nearest neighbors graph, refines it with a minimum spanning tree, and defines a graph-level trajectory $\delta_i$8 by shortest paths between low- and high-severity graphs (Shi et al., 10 Feb 2025). In clinical trajectory analysis based on elastic principal graphs, pseudotime is the geodesic distance from a chosen root node to the patient’s projection on a principal tree (Golovenkin et al., 2020). In stage-aware Bayesian networks for Alzheimer’s disease, latent stage $\delta_i$9 is estimated from baseline biomarker profiles and used to parameterize spline-varying structural equations for future annualized regional tau-PET change (Le, 14 Jun 2026).

3. Estimation regimes and data structures

Latent disease pseudotime is used in both longitudinal and cross-sectional regimes. In longitudinal EHR modeling, the input may be an $d_{ij}^k \in [0,1]$0 matrix ordered by date, with masking and interpolation for missingness, as in the cancer deep state-space model trained on laboratory tests, vitals, height/weight, and sex (Nakamura et al., 2023). In Bayesian progression forecasting, irregular sampling is handled by placing actual visit ages $d_{ij}^k \in [0,1]$1 directly into the latent sigmoid, while missing target values are omitted from the ELBO (Zhu et al., 2018). In systemic sclerosis, irregular clinical histories are modeled with a latent temporal process conditioned on time points, medications, and demographics, and inference is performed from partial histories $d_{ij}^k \in [0,1]$2 (Trottet et al., 2024).

These longitudinal formulations are complemented by methods that recover pseudotime from static or partially observed snapshots. Graph-level Pseudotime Analysis is motivated by the fact that transcriptomic disease studies often provide only cross-sectional snapshots from different subjects; it transfers the logic of pseudotime to subject-level pathway graphs so that disease progression can be reconstructed when repeated within-subject sampling is unavailable (Shi et al., 10 Feb 2025). In pathology representation analysis, diffusion pseudotime is applied to histopathology patch embeddings, with the medoid of the earliest disease class used as the root patch and pseudotime defined as the diffusion pseudotime distance from that root (Vig et al., 29 Jan 2026). In large synchronic clinical datasets, elastic principal trees similarly treat the data cloud as a geometric sampling of hidden disease routes and define progression as geodesic distance from a quasi-normal or least severe root state (Golovenkin et al., 2020).

Anchoring and direction assignment are therefore central practical issues. Some models anchor disease time to clinical diagnosis or latent onset (Lespinasse et al., 2023). Some choose roots from the earliest class, least severe state, or clinically quasi-normal node (Vig et al., 29 Jan 2026, Golovenkin et al., 2020). Some infer start and end graphs from severity extrema on a minimum spanning tree (Shi et al., 10 Feb 2025). This suggests that latent disease pseudotime is not only a statistical object but also a directional convention, and that the clinical meaning of early versus late stages depends on how the latent axis is oriented.

The same point holds for discrete latent-stage models. In the time-dependent probabilistic generative model for disease progression, the latent stages $d_{ij}^k \in [0,1]$3 are constrained to be non-decreasing, which makes the stage index a discrete pseudotime. In the continuous-time hidden Markov model with auxiliary surrogate labels, the latent state process $d_{ij}^k \in [0,1]$4 provides an ordered disease trajectory even though the model does not produce a single continuous pseudotime scalar (Zaballa et al., 2023, Cai et al., 2024).

4. Interpretability, staging, and mechanistic readouts

A central feature of latent disease pseudotime is that it is usually introduced not merely to improve prediction, but to make disease progression interpretable. In the deep state-space analysis framework, interpretability comes from a full pipeline: latent-state estimation, UMAP visualization, $d_{ij}^k \in [0,1]$5-means clustering, inter-cluster transition probabilities, and examination of abnormal laboratory values at each stage. The authors report that three clusters, chosen based on silhouette score and interpretability, correspond to a dangerous/near-death state, an intermediate state, and a stable state; final latent states of deceased and surviving patients are clearly separated; and patient trajectories predominantly move from cluster (1) to (2) to (3), with very few direct transitions from (1) to (3) (Nakamura et al., 2023).

Other frameworks make the latent axis interpretable by linking it to biomarker orderings. In a probabilistic Alzheimer’s disease progression model, the inferred ordering of target inflection points places MMSE earliest, then ADAS-Cog, then CDR-SB, with MMSE about 3.5 years earlier than ADAS-Cog and ADAS-Cog about 11 years earlier than CDR-SB (Zhu et al., 2018). In diagnosis-anchored dementia modeling, the estimated latent disease time spans over twenty years before clinical diagnosis, and, for a woman aged 70 with a high level of education and APOE4 carrier status, CSF markers of tau proteins accumulation preceded markers of brain atrophy by 5 years and cognitive decline by 10 years (Lespinasse et al., 2023).

Several papers extend interpretability into explicitly mechanistic territory. Graph-level Pseudotime Analysis uses graph regression and Jacobian sensitivity to identify sensitive pathways and stage-transition pathways, then fits neural stochastic differential equations to define pathway stability and bifurcation / critical transition points, including a “point of no return” around Step 4 in the JR5558 retinal degeneration trajectory (Shi et al., 10 Feb 2025). Stage-aware Bayesian networks with latent time embedding use posterior spline-varying structural equations and g-formula contrasts to identify a mid-pseudotime window of amyloid sensitivity for future tau accumulation (Le, 14 Jun 2026). Dynamic causal discovery with latent pseudotime similarly treats causal effects as functions of disease stage and reports stage-dependent biomarker relationships such as a constant effect of age on GFAP and a strongest-early effect of pTau on NfL (Glazman et al., 6 Nov 2025).

Semi-supervision is another route to interpretability. In systemic sclerosis, guidance networks map organ-specific medical labels to disjoint subsets of the latent dimensions, so that lung, heart, and joint involvement and stage are aligned with distinct parts of the latent temporal process. The latent trajectory is then interpretable as movement through regions with increasing probabilities of organ involvement rather than as an uninterpreted embedding (Trottet et al., 2024).

5. Disease domains and empirical use cases

The disease domains covered by latent disease pseudotime are broad. In oncology EHR analysis, time-series data from 12,695 cancer patients were used to estimate latent states related to prognosis, visualize temporal transitions, and identify anticancer-drug-specific laboratory signatures such as low lymphocytes and high segmented neutrophils for nivolumab and low CK with high LDH and BUN for bicalutamide (Nakamura et al., 2023). In retinal degeneration, subject-level pathway graphs derived from 24,888 genes mapped to 343 canonical molecular pathways support graph-level disease trajectories, sensitive-pathway analysis, and stability or bifurcation analysis (Shi et al., 10 Feb 2025).

Neurodegenerative disease is the most extensively developed application area. Alzheimer’s disease progression has been modeled through sigmoidal latent disease states personalized by APOE, sex, education, and MRI features (Zhu et al., 2018); through diagnosis-anchored latent disease time in a multivariate mixed model over 12 ADRD markers (Lespinasse et al., 2023); through latent time joint mixed-effects models for multicohort ADNI data (Li et al., 2017); through latent processes for cerebral anatomy, cognitive ability, and functional autonomy linked by temporal influence matrices (Taddé et al., 2018); through dynamic causal discovery over a latent pseudotime inferred from cross-sectional AD data (Glazman et al., 6 Nov 2025); and through stage-aware Bayesian networks that forecast regional tau progression while constraining dependencies according to biologically plausible AT(N) ordering (Le, 14 Jun 2026). In ALS, latent disease age has been coupled to a Weibull survival model so that longitudinal decline and event risk are aligned on the same biological axis (Ortholand et al., 2024).

Systemic sclerosis provides a contrasting use case in which the latent object is not a scalar but a structured latent disease manifold. The semi-supervised deep generative model for SSc learns temporal latent processes from irregular longitudinal data, uses dynamic time warping on latent trajectories for similarity and clustering, and reports three clusters interpreted as mild, medium, and high severity trajectories. Predictive clustering achieves a macro-$d_{ij}^k \in [0,1]$6 of $d_{ij}^k \in [0,1]$7 for early assignment to severity clusters (Trottet et al., 2024).

Cross-sectional and representation-level applications extend the concept beyond classical longitudinal medicine. Diffusion pseudotime applied to pathology foundation model embeddings shows that all pathology-specific models recover trajectory orderings significantly exceeding null baselines across four validated cancer progressions, with vision-only models achieving the highest fidelities and $d_{ij}^k \in [0,1]$8 on CRC-Serrated (Vig et al., 29 Jan 2026). Elastic principal trees applied to myocardial infarction and diabetes datasets recover branching clinical trajectories, terminal states, and pseudotime-stratified hazards in large mixed-type clinical data (Golovenkin et al., 2020). Continuous-time hidden Markov models with surrogate labels use ordered latent disease states to distinguish AD from related dementias such as LBD when gold-standard neurological diagnoses are unavailable during life (Cai et al., 2024).

6. Limitations, caveats, and common misconceptions

A common misconception is that latent disease pseudotime is equivalent to chronological age or to observed follow-up time. The recent Alzheimer’s disease causal-discovery study directly compared the two and reported pseudotime AUC $d_{ij}^k \in [0,1]$9 versus age AUC $z_{1:T}$0 for diagnosis prediction, illustrating that age is a broad risk factor whereas pseudotime is intended to represent disease stage (Glazman et al., 6 Nov 2025). Related models also separate age effects from disease progression by including both observed age and latent disease time in the same formulation (Li et al., 2017).

A second misconception is that pseudotime must be a single monotone scalar. Some methods indeed impose monotonicity, as with sigmoidal latent disease states or non-decreasing stage sequences (Zhu et al., 2018, Zaballa et al., 2023). Others do not. The cancer deep state-space model explicitly states that the latent states are not explicitly constructed as pseudotime and that the model does not impose a monotonic single-axis ordering, even though the learned latent trajectories behave like a disease progression embedding (Nakamura et al., 2023). The systemic sclerosis models likewise define progression through multivariate latent trajectories rather than a single scalar pseudotime (Trottet et al., 2024, Trottet et al., 2023). A plausible implication is that “latent disease pseudotime” is best understood as a family resemblance term covering scalar, discrete, and manifold-valued progression representations.

A third caveat concerns causality and identifiability. Retrospective EHR analyses cannot establish causality between latent-state transitions and outcomes or laboratory abnormalities, and prospective validation is still needed (Nakamura et al., 2023). Stage-aware causal models require structural assumptions: one AD framework assumes faithfulness and causal sufficiency when incorporating latent pseudotime, while another constrains directions according to a biologically plausible AT(N) layer ordering (Glazman et al., 6 Nov 2025, Le, 14 Jun 2026). These approaches are mechanistically interpretable, but they do not remove the dependence on modeling assumptions.

Hyperparameters, preprocessing, and anchors materially affect the estimated progression axis. The deep state-space EHR model reports that estimation results vary with hyperparameters and that $z_{1:T}$1 and learning rate $z_{1:T}$2 were selected as easiest to interpret; its three-cluster staging is clinically useful but still a simplification of heterogeneous cancer trajectories (Nakamura et al., 2023). The diagnosis-anchored dementia model shows that a non-anchored latent-time model can fit slightly better in RMSE while producing latent times for incident dementia cases that are often far from actual diagnosis times, indicating that good statistical fit alone does not guarantee meaningful staging (Lespinasse et al., 2023). Principal-tree approaches require a clinically sensible root state because the latent trajectory has no intrinsic direction (Golovenkin et al., 2020). In systemic sclerosis, interpretability depends on sparse medical labels and on preliminary hand-crafted label definitions (Trottet et al., 2024).

Taken together, these caveats indicate that latent disease pseudotime is neither a single algorithm nor a guaranteed biological truth. It is a modeling strategy for reconstructing disease-stage structure from incomplete observations. Its scientific value depends on how well the latent axis aligns with prognosis, biomarker ordering, mechanistic hypotheses, or clinically meaningful transitions in the specific disease setting under study.