Typical Healthcare Pathways as a Basis for Admixture Modeling of Patient Trajectories

Abstract: Background: Understanding whether patients follow similar or distinct patterns of care is important for characterizing clinical practice, identifying patient subgroups, and supporting quality improvement. However, routine healthcare trajectories are difficult to compare directly because patients may differ in their diagnostic workup, treatment sequencing, timing of clinical events, and documentation practices. Despite this variation, trajectories often contain recurring patterns at the cohort level. Methods: To address this challenge, we present a framework that explicitly separates cohort-level typical pathway identification from patient-level inference. At the cohort level, we derive an interpretable representation of care processes using a rule-based algorithm to identify typical healthcare pathways, resulting in a compact pathway graph. These pathways are then modeled as Markov chains and used as structured components in an admixture model, allowing each patient to be represented as a probabilistic mixture of typical pathways rather than being assigned to a single pathway component. The resulting admixture weights provide a compact representation of patient trajectories for subgroup characterization. We further assess the stability of the identified pathways and inferred admixture representations across multiple train-test splits. Results: Across train-test splits, the framework demonstrated consistent pathway structures and patient-level mixture patterns. Applied to routine care data from prostate cancer patients undergoing radical prostatectomy, the framework identified interpretable care patterns and supported the identification of patient subgroups with similar clinical event patterns. Conclusion: Overall, the proposed framework provides an interpretable and stable approach for summarizing treatment pathways and characterizing patient subgroups in real-world practice.

• The paper introduces a two-stage framework that simplifies EHR trajectories into typical pathways and estimates each patient’s mixed membership using fixed first-order Markov models.
• The method identified three stable, clinically interpretable pathways in 995 prostatectomy patients, with SLSQP outperforming EM in clustering quality, achieving a 0.751 silhouette score versus 0.698.
• Simulation results show robust mixture recovery under increasing trajectory noise, while limitations include threshold-dependent pathway extraction, first-order dependencies, and limited validation beyond one institution.

Motivation and positioning

Routine care trajectories recorded in electronic health records (EHRs) are heterogeneous: patients with the same diagnosis differ in diagnostic workup, treatment sequencing, event timing, and documentation practices, producing fragmented, highly branched event logs even when the underlying clinical process is standardized. The paper by Farhadizadeh et al. addresses the problem of deriving compact, interpretable cohort-level representations of such trajectories while retaining a principled probabilistic account of individual patients. The authors argue that existing approaches each solve only part of this problem: process mining captures care flows but retains substantial patient-level variability; topic models reduce dimensionality but treat structural progression only implicitly; clustering approaches partition heterogeneity without a unified probabilistic representation; and predictive deep learning methods target outcomes rather than interpretable care progression. The proposed framework explicitly separates cohort-level pathway identification from patient-level inference, combining a rule-based pathway simplification algorithm with an admixture (mixed-membership) model borrowed conceptually from population genetics (2606.14587).

Clinical data

The framework is illustrated on routinely collected inpatient coding data from 995 prostate cancer patients undergoing radical prostatectomy at the University Medical Center Freiburg between 2015 and 2020, processed within the German Medical Informatics Initiative. The cohort comprises 6,904 inpatient events encoded as ICD-10 admission diagnoses (4 distinct codes), OPS procedure codes (162 distinct codes across six catalog chapters), and discharge markers, yielding 167 distinct state labels after numeric encoding. Substantial structural fragmentation is documented: trajectories vary in length depending on whether diagnostics were completed externally or in-house, and concurrent coding practices (e.g., robotic-assisted prostatectomy coded simultaneously as OPS-5.604 and OPS-5.98) generate partially overlapping event sets. This dataset motivates the central design requirement that any cohort-level summary must tolerate documentation-driven variability without discarding clinically meaningful transitions.

Step 1: Cohort-level typical pathway identification

The first stage constructs a prefix tree from temporally ordered patient trajectories, where nodes represent event blocks (sets of codes at one timestamp) annotated with patient counts and identifiers. A key methodological contribution is an importance score for nodes and procedures designed to identify structurally informative branching points. For node $N_i$ within a care stage, importance is defined as $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$, where $p_i$ is the relative frequency within the stage. This assigns maximal weight to nodes splitting the cohort into comparably sized subgroups, down-weighting both rare deviations and near-universal routine events; procedure importance aggregates node scores over all nodes containing the procedure.

Simplification then proceeds through four operations guided by a user-specified importance threshold $\tau$: vertical collapsing merges same-stage nodes sharing an important procedural subset; horizontal collapsing compresses chains of non-important nodes into composite nodes; pruning removes nodes failing absolute ($\alpha$) and relative ($\beta$) support criteria; and backward merging with end-node collapsing consolidates structurally equivalent nodes missed by the forward pass. The threshold parameters control the resolution–interpretability trade-off rather than being inferred from data — a design choice the authors acknowledge as descriptive rather than model-based.

Applied to the prostate cancer cohort with $\tau = 15$, $\alpha = 5$, $\beta = 0.1$, the algorithm produced a graph whose backbone runs from diagnostic evaluation through biopsy confirmation to surgery and discharge, with side branches reflecting referral-status variation. Notably, 29.8% of original patient trajectories were fully retained after collapsing and pruning, indicating that the simplification preserves complete pathways for roughly three in ten patients while compressing the remainder. Sensitivity analyses showed that varying thresholds affected mainly peripheral branches and granularity, not pathway identity, and the same three pathway structures were recovered across random train–test splits.

Step 2: Admixture inference over fixed pathway components

Each identified typical pathway is converted into a first-order Markov chain over the common state space, with transition matrices $P_k$ estimated from empirical transition counts along the pathway subgraph and held fixed thereafter. Patient-level inference then follows the mixed-membership paradigm of STRUCTURE and ADMIXTURE (2606.14587): admixture weights $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$0 on the $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$1-simplex are estimated by maximizing

$$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$2

Two optimizers are compared: an EM procedure introducing latent per-transition pathway assignments, and direct constrained optimization via SLSQP. Because the $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$3 are fixed and derived from training data only, test-set inference avoids information leakage, and the strict 80/20 train–test separation ensures the "typical" pathways serve as a fixed scaffold.

A first-order Markov assumption is a substantive modeling restriction: longer-range dependencies and context effects spanning multiple care steps are not captured. The authors justify it on parsimony and stability grounds given the simplified pathway representation, but this remains an approximation to real-world trajectory dynamics.

Step 3: Clustering in admixture space

Admixture vectors define a shared low-dimensional space in which patients are clustered with $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$4-means (with $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$5 set equal to $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$6), avoiding sequence alignment entirely. In the application, $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$7 pathways emerged: an upper pathway dominated by combined diagnostic/procedural states progressing to intervention and discharge with readmission transitions; a middle pathway with additional intermediate stages; and a lower pathway involving discharge followed by readmission for further procedures before final discharge.

Clustering validation favored SLSQP over EM consistently:

Optimizer	Silhouette	Calinski–Harabasz	Davies–Bouldin
EM	0.698 ± 0.030	562 ± 64	0.436 ± 0.070
SLSQP	0.751 ± 0.016	727 ± 50	0.366 ± 0.044

Across all splits, test log-likelihoods were uniformly higher under SLSQP (e.g., −1808.51 vs. −1818.68), indicating both better fit to the admixture objective and stronger cluster separation. The three clusters mapped onto clinically interpretable patterns: extensive in-hospital workup including fusion biopsy, direct referral-to-prostatectomy trajectories with external staging, and mixed contributions suggesting more complex perioperative courses. This interpretability is the practical payoff of the admixture formulation — intermediate mixture weights correspond to genuinely blended care patterns rather than forced single-pathway assignments.

Simulation evidence

A supplementary simulation study with $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$8 patients and three partially overlapping latent backbones evaluated recovery under increasing structural mixing (Dirichlet concentration $$\text{Imp}(N_i) = \min(p_i, 1-p_i) \cdot \sum_{j \in \mathcal{C}_i} c_j$$9 from 0.8 to 1.2; switching probability from 0.10 to 0.35), with realistic perturbations including node skipping, repetition, vertical and horizontal noise. Continuous mixture recovery degraded moderately (Pearson correlation 0.834 → 0.754) and dominant-backbone recovery (ARI) declined monotonically from 0.940 to 0.717, while silhouette scores remained above 0.87 throughout. The authors emphasize two points: performance did not collapse under high mixing, demonstrating robustness, and it did degrade monotonically, confirming the task was non-trivial. MAE stabilized around 0.24 partly because higher Dirichlet concentration produces more diffuse ground-truth weights, which mechanically limits absolute error growth — a caveat worth noting when interpreting the MAE column.

Limitations and open questions

The authors are explicit about several constraints. The pathway algorithm is descriptive and threshold-dependent; although sensitivity analyses showed pathway identity was stable, the choice of $p_i$0, $p_i$1, $p_i$2 is ultimately user-specified and no principled selection criterion is provided. The first-order Markov admixture model does not capture longer-range temporal dependencies, a trade-off against richer but less interpretable sequential deep learning models. The analysis is retrospective on completed hospital trajectories ending in discharge, so the framework characterizes care after the fact and cannot support real-time decision-making without further development. Generalizability across institutions, coding practices, and patient populations remains untested, and the clinical meaningfulness of admixture-derived subgroups — while plausible here — has not been validated against outcomes. An open question the paper leaves unresolved is whether the flat-likelihood and non-uniqueness issues known to affect admixture estimation in population genetics also arise in the healthcare setting, and how they would affect subgroup interpretation.

Conclusion

This paper contributes a structured pipeline that couples rule-based, importance-guided compression of prefix trees into typical healthcare pathways with a fixed-component admixture model for patient-level trajectory representation. On a single-center prostatectomy cohort, it yielded stable, clinically interpretable pathway structures, well-separated patient subgroups (Silhouette up to 0.751 under SLSQP), and simulation-verified recovery of latent mixtures under controlled structural ambiguity. Its principal strengths are interpretability, reliance on widely available billing-coded data, and demonstrated train–test stability; its principal restrictions are the descriptive nature of pathway extraction, the first-order Markov assumption, and retrospective scope.