NT-proBNP and Progression in Pulmonary Hypertension

Why Model NT-proBNP and Progression Together?

In pulmonary hypertension, NT-proBNP is measured repeatedly, while clinical worsening or death may occur at any time. NT-proBNP can be strongly right-skewed, and a patient’s unobserved underlying trajectory is not identical to each noisy assay result. Moreover, a worsening event can stop further measurements. An analysis that treats the observed biomarker values as if they were error-free, or ignores why later measurements are missing, can therefore give a misleading account of the biomarker–outcome association.

A joint model represents the repeated measurements and event time in a linked framework. Here the event is a composite of clinical worsening or death, with time measured from a defined study origin. Baseline functional status, such as New York Heart Association (NYHA) class, is clinically relevant: it may predict the event and may also modify how strongly the underlying NT-proBNP level is associated with the event hazard. These are distinct roles and should be modeled and interpreted separately.

A Joint Model for Log NT-proBNP and Event Time

Let \(Y_{ij}\) be the NT-proBNP concentration for patient \(i\) at visit time \(t_{ij}\), measured in pg/mL. A common analysis models the natural logarithm of concentration. Let \(F_i=1\) indicate worse baseline functional status, for example NYHA class III/IV, and \(F_i=0\) indicate class I/II. The continuous underlying log NT-proBNP trajectory is \(m_i(t)\); it is latent because measurements contain error.

Here \(b_i=(b_{0i},b_{1i})^\mathsf{T}\) are patient-specific random effects, commonly modeled as multivariate normal; \(\epsilon_{ij}\) are measurement errors, commonly independent normal errors conditional on the random effects. The baseline hazard is \(\lambda_0(t)\), and \(\gamma\) represents the functional-status association at \(m_i(t)=0\). The coefficient \(\alpha\) is the log-hazard association per one-unit increase in latent log NT-proBNP for the reference status group. The interaction \(\delta\) allows that association to differ for the worse-status group, whose association coefficient is \(\alpha+\delta\). Time and the functional-status terms in \(m_i(t)\) allow the average biomarker trajectory to vary by status.

The stated distributional and structural assumptions are modeling choices, not universal facts. They should be checked against the study’s measurement schedule, observed trajectories, event pattern, and clinical context. In particular, a linear trajectory on the log scale may be inadequate if the biomarker rises nonlinearly. The model also assumes that, conditional on the specified covariates and shared random effects, the longitudinal errors and event process are appropriately represented and residual measurement error does not itself carry further event information.

What the Association Means for Clinical Worsening or Death

The survival submodel relates the instantaneous event rate among patients still event-free at time \(t\) to the current latent log NT-proBNP value and baseline functional status. It does not assert that NT-proBNP causes worsening or death. Other disease processes may influence both the biomarker and the event, and the association is conditional on the model and included covariates.

For two otherwise comparable patients in the same functional-status group, at the same time, with latent log NT-proBNP values differing by \(d\), the modeled hazard ratio is \(\exp\{(\alpha+\delta F_i)d\}\). Because a doubling in concentration is an increase of \(\log(2)\) on the natural-log scale, its hazard ratio follows directly.

This is a conditional hazard ratio, not a risk ratio or a ratio of survival probabilities. It compares instantaneous hazards at a given time among patients with the same modeled covariates and status, differing in latent concentration by a factor of two. It also relies on the proportional-hazards form: the coefficient-based relative hazard is constant over time unless the model explicitly allows a time-varying association.

Justification Through the Joint Likelihood

The reason to link the two submodels is that event-free follow-up determines which longitudinal measurements can be observed. Write \(T_i\) for event time, \(C_i\) for censoring time, \(\tilde T_i=\min(T_i,C_i)\), and \(\Delta_i=1\) if the event is observed and 0 if censored. Conditional on random effects \(b_i\), the event contribution has the familiar hazard form. The joint likelihood integrates the shared random effects rather than pretending the latent trajectory is observed exactly.

This expression follows by multiplying the conditional density of the observed longitudinal measurements by the conditional event-time contribution and the random-effects density, then integrating over the unobserved \(b_i\). In the event term, \(\lambda_i(\tilde T_i\mid b_i)^{\Delta_i}\) contributes the hazard for an observed event and contributes 1 for censoring; the exponential is the probability of remaining event-free through the observed follow-up time, conditional on \(b_i\). The integration accounts for uncertainty about each patient’s trajectory and links it to the event process. The usual conditional independent-censoring assumption is needed: given the modeled information, censoring must not carry additional information about the event time.

If a patient’s event stops biomarker collection, the number and timing of measurements can depend on the event process. A joint likelihood can account for this connection through shared random effects and the modeled latent trajectory, subject to its assumptions. It does not automatically solve every missing-data problem: informative visit scheduling, discontinuation for unmodeled reasons, or misspecified random-effects distributions may require additional modeling.

Diagnostics, Alternatives, and Limits

Diagnostics should assess both components and their connection. Plot observed log NT-proBNP trajectories over time, separated by baseline functional status, and compare their broad shape with fitted trajectories. Examine residual patterns for trends, changing variance, outliers, or inadequately modeled curvature. For the event component, assess whether the assumed baseline hazard and proportional association are plausible, and check whether residual or predictive summaries reveal systematic lack of fit. Sparse measurement near the event, few events in a status group, or poor overlap in biomarker values across groups can make an interaction estimate especially unstable. Convergence problems, extreme random-effect estimates, or sensitivity to trajectory specification are warnings, not merely software inconveniences.

The joint model is not the only option. A time-dependent Cox model that inserts each latest observed NT-proBNP result can be simpler, but the last observed result is measured with error and may be stale; treating it as the current latent value can bias the association. Landmark analysis defines a prediction time and uses information available by that time, making its target explicit, but it does not use the same full longitudinal-event likelihood and typically answers a prediction question conditional on reaching the landmark. A baseline-only survival model is straightforward but discards subsequent biomarker evolution. These approaches can be useful for different estimands; their results should not be assumed interchangeable.

A joint model remains conditional on choices about transformation, trajectory shape, random effects, association structure, and censoring. An \(\alpha\) estimate is not evidence of a causal effect, nor does it alone establish that NT-proBNP improves clinical prediction. Clinical utility requires appropriate validation and decision-focused evaluation. The model also does not imply that functional class is measured without error or that the same association applies outside the population and follow-up represented in the data.

Common Pitfall