FVC Decline and Progression in Idiopathic Pulmonary Fibrosis

Why FVC and Progression Belong in One Analysis

Idiopathic pulmonary fibrosis (IPF) is a progressive lung disease. Forced vital capacity (FVC), commonly measured in liters, is recorded repeatedly, while patients may later die or receive a lung transplant. The scientific question is not only whether FVC tends to decline, but also how a patient’s evolving lung function and baseline disease severity relate to the timing of this clinical event.

A central complication is that FVC measurements often stop when a patient dies or receives a transplant. Patients with rapid deterioration may therefore contribute fewer measurements than patients who remain event-free. An analysis that treats all observed FVC values as if their availability were unrelated to the event can misrepresent the underlying trajectories and their association with progression.

A Joint Model for Trajectory and Event Risk

One useful longitudinal specification is a linear mixed-effects model. It allows average FVC change to differ with baseline severity and allows each patient to have a patient-specific intercept and slope. The latent, or underlying, FVC trajectory is the expected measurement after accounting for measurement error.

Here \(\epsilon_{ij}\) is measurement error, commonly assumed to be independent normal error with mean zero and variance \(\sigma^2\), conditional on the patient-specific random effects. The random effects \((b_{0i},b_{1i})\) are commonly modeled jointly as mean-zero normal variables with covariance matrix \(D\). They represent variation across patients in baseline FVC and rate of change. Under this model, the patient’s latent rate of FVC change is \(m_i'(t)=\beta_1+\beta_3S_i+b_{1i}\), in liters per year.

A shared-parameter event model connects the latent trajectory to event risk. For example, a current-value association model is

The baseline hazard \(\lambda_0(t)\) describes the hazard over time for the reference values of the covariates; it may be left unspecified. The coefficient \(\gamma\) is the log hazard ratio per one-unit increase in baseline severity, holding the latent FVC trajectory fixed. The coefficient \(\alpha\) is the log hazard ratio per one-liter increase in current underlying FVC, holding baseline severity fixed. Since higher FVC would generally be expected to correspond to lower risk, a negative \(\alpha\) is consistent with that direction. It is an estimated association, not proof that changing FVC would change an individual’s outcome.

The model may instead, or additionally, associate risk with the rate of decline:

Because a faster decline has a more negative slope, an association in which faster decline corresponds to higher hazard would have \(\eta<0\). Current value and slope represent different scientific questions. Including both requires enough data and adequate precision to distinguish their effects; it should not be an automatic modeling choice.

How the Joint Likelihood Handles Event-Related Missing Measurements

Conditional on the random effects, the longitudinal measurements contribute their measurement densities, while the event record contributes a survival likelihood. If \(f_y\) denotes the longitudinal density and \(S_i(t\mid b_i)\) the conditional event-free survival function, the patient’s contribution is obtained by integrating over the random effects:

For a censored patient, \(\delta_i=0\), so the event contribution is \(S_i(T_i\mid b_i)\); for a patient with an observed event, it is \(\lambda_i(T_i\mid b_i)S_i(T_i\mid b_i)\). The same random effects inform both parts. Thus, a patient’s observed FVC pattern helps estimate the underlying trajectory, and that trajectory informs the event risk even when measurements stop at the event.

This does not mean that the model creates FVC observations after death or transplant. Rather, it uses the joint probability of the measurements and event to estimate the association under its assumptions. If patients with worsening unmeasured health stop attending visits before the recorded event, the assumed observation process may be inadequate.

Diagnostics, Alternatives, and Limitations

The joint model is only as credible as its component assumptions. Examine plots of observed FVC against time, both overall and by baseline severity, and compare them with fitted population and patient-specific trajectories. Check residual patterns for nonlinearity, changing variance, and influential observations. A linear time effect may be inadequate if decline is curved; a flexible time function may be needed. Inspect random-effects estimates and their uncertainty, especially when patients have few measurements.

For the event component, assess whether the assumed proportional-hazards structure is plausible and whether the baseline hazard is sufficiently flexible. Examine uncertainty and correlation among \(\alpha\), \(\eta\), and \(\gamma\) if several associations are fitted. A large standard error or unstable estimates can signal that the data do not distinguish current FVC, slope, and baseline severity. Compare predicted and observed event patterns where appropriate, and conduct sensitivity analyses for plausible alternative trajectory and association structures.

A separate mixed model for FVC and survival model for the event is simpler, but it does not by itself account for the dependence induced when event risk and measurement availability share a latent trajectory. A two-stage approach that first estimates each patient’s slope and then uses it as a covariate in a survival model can ignore uncertainty in those estimated slopes and may perform poorly when measurements are sparse. A time-updated analysis using observed FVC is another alternative, but it may not represent the underlying value and can be vulnerable to irregular measurement and event-related dropout. These approaches can still be useful for different estimands or when their assumptions are justified; they are not interchangeable with a joint model.

Limitations remain even when the joint model fits well. The association is conditional on measured covariates and the specified model, and unmeasured confounding may remain. FVC depends on measurement quality and patient effort; missing visits may reflect changing health not captured by the model. A composite death-or-transplant endpoint combines clinically different events, so its association should not be interpreted as an effect on either component separately. Finally, extrapolating a patient’s latent trajectory beyond the observed time range depends strongly on the assumed trajectory form.