Estimation of the distribution of longitudinal biomarker trajectories prior to disease progression

Abstract

Most studies characterize longitudinal biomarker trajectories by looking forward at them from a commonly used time origin, such as the initial treatment time. For a better understanding of the relationship between biomarkers and disease progression, we propose to align all subjects by using their disease progression time as the origin and then looking backward at the biomarker distributions prior to that event. We demonstrate that such backward-looking plots are much more informative than forward-looking plots when the research goal is to understand the shape of the trajectory leading up to the event of interest. Such backward-looking plotting is an easy task if disease progression is observed for all the subjects. However, when these events are censored for a significant proportion of subjects in the study cohort, their time origins cannot be identified, and the task of aligning them cannot be performed. We propose a new method to tackle this problem by considering the distributions of longitudinal biomarker data conditional on the failure time. We use landmark analysis models to estimate these distributions. Compared to a naïve method, our new method greatly reduces estimation bias. We apply our method to a study for chronic myeloid leukemia patients whose BCR-ABL transcript expression levels after treatment are good indicators of residual disease. Our proposed method provides a good visualization tool for longitudinal biomarker studies for the early detection of disease.

Keywords: biomarker; disease recurrence; landmark analysis; survival analysis.

FIGURE 1

BCR-ABL expression level trajectories showing down-and-up patterns for CML patients receiving treatment.

FIGURE 2

Top: Distribution of BCR-ABL expression levels over time for all subjects, using the beginning of treatment as the time origin. Bottom: Distribution of BCR-ABL expression levels over time for all subjects for whom the time of disease progression was observed, using disease progression as the time origin.

FIGURE 3

Estimated mean biomarker levels prior to disease progression, by the proposed method (dashed line) and a naïve method that ignores censored subjects (dotted line), compared with the true values (solid line): top panels for scenario 1 (constant regression coefficient β); bottom panels for scenario 2 (time varying coefficient β(t) = β log(t); left panels, 50% censoring rate; right panels, 15% censoring rate.

FIGURE 4

Quantile distributions of biomarker levels prior to disease progression for the scenario 1 (constant regression coefficient β over time). Left column panels: Truth; Middle column panels: by the proposed method; Right column panels: by the naïve method ignoring censored subjects. Top row panels: 50% censoring rate; bottom row panels: 15% censoring rate.

FIGURE 5

Estimated mean (solid lines) and 75% quantile (dashed lines) of BCR-ABL transcript levels of CML patients prior to disease progression, grouped by age (< 60 or ≥ 60 years) and treatment dose level (high or low), by the proposed method (black lines) and the IPCW method (gray lines), respectively.

FIGURE 6

Estimated mean biomarker levels prior to disease progression, by the proposed method (dashed line) and a naïve method that ignores censored subjects (dotted line), compared with the true values (solid line)

FIGURE 7

The IPCW method by Chan and Wang gives biased results. Legend: true values shown by solid line, IPCW estimates by dashed line, 95% confidence intervals by dotted lines.

FIGURE 8

Top panels: The asymptotic 95% confidence intervals (CIs) and the empirical 95% CIs by the proposed method (all averaged over 1,000 simulations). Bottom: The coverage probabilities of the 95% CIs by the proposed method (dotted line: 95%). Legend for CIs: true values shown by solid line, estimates by dashed line, 95% CIs by dotted lines.