Biomarker Trajectories In Disease Progression

Why Model the Trajectory?

A biomarker measured repeatedly during a trial contains two kinds of change: the disease process that would occur without the investigational treatment, and any change associated with treatment. The control group helps estimate the first component. Repeated measurements then show whether the treated group follows a different path, rather than merely ending at a different value. This distinction matters when a biomarker naturally worsens, improves, or approaches a plateau over time.

A fitted trajectory is useful only if its assumptions are plausible and its predictions agree with the observed data over the trial duration. A model that fits an average endpoint but misses the intervening measurements may misrepresent when a treatment effect appears or how it develops.

A Basic Disease-Progression Model

Let \(Y_{ij}\) be the biomarker value for participant \(i\) at time \(t_{ij}\), measured in months from the baseline assessment. Let \(D_i=0\) for control and \(D_i=1\) for treatment. Suppose higher biomarker values indicate worse disease. A simple linear model is:

Here, \(B_i\) is participant \(i\)'s expected baseline value, \(k_0\) is the control-group progression rate, and \(\delta\) is the treatment-associated difference in progression rate. Both rates have units of biomarker units per month. The expected control trajectory is \(B_i+k_0t\); the expected treated trajectory is \(B_i+(k_0+\delta)t\). When higher values are worse, a negative \(\delta\) indicates slower worsening under treatment, provided the model and trial design support that interpretation.

The model assumes a linear expected trajectory over the interval being modeled; a consistent measurement scale; appropriate handling of baseline differences; and residual errors with mean zero conditional on the model. In the simplest version, errors are independent and have constant variance. Repeated measurements from the same participant are generally correlated, so an individual-level analysis may instead include participant-specific random effects and an appropriate residual covariance structure. Such choices should reflect the sampling schedule and data, not be added mechanically.

If treatment is expected to produce an immediate biomarker shift as well as a progression-rate change, the model can include a treatment-associated offset after treatment begins. These effects have different trajectory signatures: an offset produces a roughly constant gap, whereas a slope difference produces a gap that changes with time. The baseline observation is usually taken before treatment, so it should not be interpreted as an effect of treatment.

What the Control Comparison Identifies

With treatment and control measurements at the same times, the difference in their changes can estimate the difference in progression rates. To see why, suppose both groups have the same expected baseline value \(B\), and use the linear model. At time \(t\), the expected control change from baseline is \(k_0t\); the expected treated change is \((k_0+\delta)t\). Subtracting the control change from the treated change gives:

Dividing this difference by \(t>0\) gives \(\delta\), the difference in rates. This is a difference-in-differences argument: it compares change within each group, then compares those changes between groups. In a randomized trial, randomization supports interpreting the between-group contrast as a treatment effect, subject to adherence, missing-data, measurement, and model assumptions. A within-treated-group change alone does not make that separation: it combines natural history with any treatment effect.

Fit and Check the Trajectory Across Time

In participant-level data, model parameters are estimated from all available measurements, not just the first and last visit. A regression or nonlinear mixed-effects fit can represent participant variation in baseline or progression rate while estimating the population-level trajectory. Once fitted, calculate expected values at the scheduled visits and compare them with observed values and residuals. A residual is the observed value minus the fitted value. A residual trend over time signals that the assumed trajectory may not capture the data.

The following workflow makes the distinction between a model fit and a treatment-effect claim explicit.

The model form is a substantive assumption, not just a computational choice. A straight line can be adequate over a limited duration even when the underlying disease process is not linear indefinitely. Conversely, if the observed trajectory bends systematically, a single constant slope can conceal the changing rate.

When Progression Is Curved

Some biomarkers change rapidly early and then approach a plateau. One possible natural-history model is an asymptotic exponential trajectory:

where \(B_0\) is the baseline value, \(A\) is the total change toward the asymptote, and \(k>0\) is a rate constant with units of inverse time. A treatment may be represented as altering \(k\), \(A\), or both, but each choice expresses a different biological pattern. These parameters are not interchangeable: changing \(k\) changes how quickly the trajectory approaches its limit, whereas changing \(A\) changes the limiting amount of change. Estimates require enough informative observations to distinguish the parameters.

Diagnostics, Alternatives, and Limitations

Evaluate predictions within the trial duration before relying on them. Plot observed and fitted trajectories by treatment group, and inspect residuals against time, fitted value, and group. Residuals should not show a sustained pattern that the model has missed. Also check whether a few participants or visits dominate the apparent trend and whether variability increases or decreases over time. In a nonlinear model, predictions should be checked at the observed visits as well as between them; a plausible-looking curve is not evidence of a good fit by itself.

The main alternatives answer related but different questions. A comparison of baseline-adjusted endpoint means is simpler and can estimate a treatment contrast at a specified final visit, but does not describe the path or use the shape of intermediate observations in the same way. Comparing group means at each visit is transparent and useful diagnostically, but does not by itself provide a coherent participant-level model of repeated measurements. A linear trajectory is parsimonious and interpretable when its constant-rate assumption is tenable; a curved model can represent deceleration or a plateau but needs enough data and stronger assumptions. The appropriate choice depends on the scientific question, visit schedule, and observed trajectory.

Randomization helps separate natural history from treatment effects, but it does not repair every problem. Differential dropout can make later observed means unrepresentative, especially if leaving the study is related to disease status. Baseline imbalance can affect crude comparisons, and measurement error or inconsistent assessment procedures can distort slopes. A model may also be weakly identified when there are few visits, a short trial, or little observed curvature. Report uncertainty in the fitted rates and predictions; a point estimate alone does not show how precisely the trajectory is known.

• Mistake: “The treated group improved, so treatment caused the improvement.” The group may have improved without treatment. Compare its change with the control trajectory.
• Mistake: “The groups differ at the final visit, so the treatment changed progression.” A final-visit difference can reflect baseline imbalance or a shift rather than a rate change. Examine baseline and the trajectory over repeated visits.
• Mistake: “A linear fit is adequate because the line passes through the endpoint means.” Intermediate observations and residual patterns can reveal curvature or lack of fit. Check predictions throughout the trial.
• Mistake: “A fitted trajectory predicts future outcomes with certainty.” Predictions depend on estimated parameters, model assumptions, and the trial data. They carry uncertainty, and predictions beyond the observed duration require additional caution.