Circulating Tumor DNA and Breast Cancer Recurrence

Why Detection Limits Matter in Recurrence Studies

After treatment for breast cancer, circulating tumor DNA (ctDNA) may be measured repeatedly to study whether its trajectory is associated with later recurrence. The measurements are not always quantitative: an assay may report a value only when ctDNA exceeds a specified detection limit. A report of “below detection” is not a measured zero. It indicates that the assay did not quantify the value, while the underlying signal may be positive and lower than the limit.

This distinction matters when measurements are used to describe change over time. Replacing below-limit results with zero, or with an arbitrary small number, can distort estimated levels and slopes. It can also distort estimated associations with recurrence if the chance of a below-limit result changes over follow-up. A joint model provides a way to represent the underlying marker trajectory and relate it to recurrence time while retaining the information that a below-limit measurement falls in a known range.

Assays do not all measure ctDNA on the same scale. The examples here use a hypothetical assay reporting ctDNA concentration in copies per millilitre with a detection limit of 1 copy per millilitre. A real analysis must use the assay’s stated scale, reporting rules, and detection limits; the numbers here are illustrative, not clinical guidance.

A Longitudinal Model With Censoring

A simple longitudinal submodel represents each patient’s trajectory with a random intercept and slope:

Here \(\beta_0\) and \(\beta_1\) describe the population-average log-concentration trajectory, while \(b_{0i}\) and \(b_{1i}\) allow patient \(i\) to differ in level and slope. The random effects have covariance matrix \(D\). The residuals are assumed independent across visits conditional on the random effects, with constant variance \(\sigma^2\). Extensions may include treatment or other baseline covariates, nonlinear time trends, or a different residual structure when justified by the data.

Let \(R_{ij}=1\) when the assay quantitatively reports the value and \(R_{ij}=0\) when it reports a value below the limit. For \(R_{ij}=1\), the contribution to the conditional measurement likelihood is a normal density at the observed \(W_{ij}\). For \(R_{ij}=0\), the contribution is the probability that the latent measurement falls below the limit:

The functions \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. The second line follows because, conditional on the random effects, \(W_{ij}\) is normal with mean \(\mu_{ij}\) and standard deviation \(\sigma\), so the probability of a below-limit result is \(P(W_{ij}\leq\ell_{ij}\mid b_i)=\Phi((\ell_{ij}-\mu_{ij})/\sigma)\). If the detection limit varies by visit, the corresponding \(\ell_{ij}\) is used at each visit.

Connecting the Trajectory to Recurrence

Let \(T_i\) denote time from the chosen origin to recurrence or censoring, and let \(\delta_i=1\) if recurrence is observed and \(\delta_i=0\) if follow-up ends first. One shared-parameter model relates the recurrence hazard to the patient’s latent current ctDNA level:

Here \(h_0(t)\) is the baseline hazard, \(X_i\) is a vector of included covariates, and \(\gamma\) is their coefficient vector. The parameter \(\alpha\) describes the association between the latent current log ctDNA level and recurrence hazard, conditional on the covariates and model. For example, \(\exp(\alpha)\) is the hazard ratio associated with a one-unit increase in \(m_i(t)\), that is, a ten-fold increase in concentration on this log-base-10 scale, if other terms are held fixed. This is an association, not evidence that changing ctDNA itself would change recurrence risk.

The shared random effects connect the two parts: a patient’s underlying trajectory informs the survival model, and recurrence outcomes inform estimation of the trajectory distribution. A likelihood contribution for one patient can be written as:

The survival term is the event density when recurrence is observed and the probability of remaining recurrence-free through \(T_i\) when the observation is censored. The integral averages over the unobserved patient-specific random effects, whose density is denoted by \(\phi_2\). Estimation commonly requires numerical integration or an equivalent computational method.

This likelihood rests on substantive assumptions. The longitudinal mean, residual distribution, and random-effects distribution must adequately represent the marker data. The hazard model must appropriately describe recurrence conditional on the latent trajectory and covariates. The event and marker processes are assumed conditionally independent given the shared random effects and included covariates, apart from their connection through the model. Censoring must be conditionally non-informative for recurrence. The model also treats the assay’s reporting limit as known and assumes that a below-limit report conveys the inequality stated by the assay. If visit timing depends on symptoms or suspected recurrence, that process may itself need modeling or careful sensitivity analysis.

Deriving the Measurement Contributions

The distinction between an observed value and a below-limit report follows directly from the observation process. If a quantitative result \(w\) is reported, its likelihood contribution is the density at \(w\), because the observation is treated as a measurement on the continuous log scale. If only \(W\leq\ell\) is known, the contribution is the probability of the whole compatible range, not a density evaluated at an invented substitute value.

This calculation assumes the normal measurement model on the log scale and a known limit. It is not a universal correction for every assay. For example, if a laboratory reports interval-valued results, uses a probabilistic detection process, or has varying batch-specific limits, the observation model should reflect that reporting mechanism rather than treating every result as simple left censoring.

Interpreting a Hazard Association

A fitted value of \(\alpha\) is meaningful only with its scale and time-varying covariates stated. The following calculation illustrates a current-value association; it does not show a treatment effect or establish a clinical threshold for action.

Diagnostics, Alternatives, and Limitations

A fitted joint model should be checked against both parts of the data and the assay process. Useful checks include comparing observed and model-predicted proportions below the limit over time; examining residual patterns for quantified log measurements; and checking whether fitted trajectories capture the spread and changes in the observed measurements. For recurrence, assess whether the selected hazard structure is plausible, whether important covariates or time patterns are omitted, and whether results depend heavily on a small number of patients or events. Examine assay batch, sample handling, and changes in detection limits: apparent trajectory changes can reflect measurement procedures rather than biology.

The model’s assumptions can fail in ways that matter. A normal residual model may fit poorly when measurements are skewed or have outliers even after transformation. A straight-line time trend may miss a rapid post-treatment decline followed by a rise. The random-effects distribution may be inadequate, or the current-value association may not capture how recurrence relates to the marker. If visit frequency increases when clinicians suspect recurrence, treating visit times as fixed and non-informative may be misleading. Sensitivity analyses with reasonable alternative trajectory forms, association structures, and censoring assumptions help show how dependent conclusions are on modeling choices.

The main alternatives answer related but different questions. Replacing below-limit values with zero or a fixed fraction of the limit is simple, but treats an interval of possible values as an exact value and can bias trajectory estimates. Analyzing only quantified values discards information and may select patients or visits with higher marker levels. A time-dependent survival model using observed measurements can be easier to fit, but may be problematic when measurements are noisy, irregularly timed, or unavailable near the event; it does not by itself handle the latent trajectory and measurement error in the same way. Landmark analysis relates information available at a chosen time to subsequent recurrence, which can provide a clear prediction question at that landmark but does not use the full longitudinal-survival likelihood. These methods are not universally inferior: their suitability depends on the scientific question and data collection.

Joint modeling also has limitations. Results depend on the longitudinal and hazard specifications, and estimation may be difficult with few recurrences, sparse measurements, or many below-limit reports. A strong fitted association does not show that ctDNA causes recurrence, nor does it alone establish that a marker is useful for clinical decisions. Predictive performance, calibration, validation in relevant populations, assay reproducibility, and clinical consequences require separate assessment.