Why CA-125 Surveillance Requires Careful Event Definitions
After treatment for ovarian cancer, serial CA-125 measurements can provide information about a patient’s disease course. A joint model relates this evolving biomarker history to time until a defined event, such as clinically confirmed recurrence. But “recurrence time” is not self-defining. It might mean the date a clinician confirms recurrence, the date a prespecified biomarker rule is met, or an unobserved biological onset time. These choices can change who is counted as having an event and when that event is recorded.
Surveillance also occurs at discrete visits, not continuously. A patient may have a low CA-125 measurement at one visit and a high measurement at the next. That pair can bracket a change in measured status, but it does not by itself reveal exactly when the change occurred, or whether an earlier change happened and then reversed. A sound analysis therefore starts by stating what was measured, how recurrence was defined, and how the event time was assigned.
A Joint Model for the CA-125 Trajectory and Recurrence
Because concentrations are positive and may vary multiplicatively, an analysis may model \(y_{ij}=\log(Y_{ij})\). One illustrative longitudinal submodel is a mixed-effects model with patient-specific random effects:
Here \(m_i(t)\) is patient \(i\)’s latent mean log CA-125 at time \(t\); \(b_{0i}\) and \(b_{1i}\) allow patients to have different underlying levels and trends. The residuals are assumed to have mean zero and the specified variance conditional on the random effects and model covariates. A linear time trend is only an example: the trajectory form should be appropriate to the follow-up period and data.
For a current-value association, a proportional hazards event submodel can be written
The hazard \(h_i(t\mid b_i)\) is the instantaneous event rate at time \(t\), conditional on remaining event-free until then and on the patient’s latent effects. The baseline hazard is \(h_0(t)\), \(w_i\) denotes included covariates, and \(\alpha\) links the latent log CA-125 value to the event hazard. The joint likelihood accounts for the repeated measurements and event information while representing their dependence through shared patient-specific effects or the specified association structure.
The usual interpretation relies on a correctly specified longitudinal mean and event submodel, an appropriate association structure, and a suitable baseline hazard. The event definition must be applied consistently, covariates and time origins must be specified clearly, and censoring must be handled under assumptions appropriate to the analysis. In particular, censoring that depends on unmodeled disease status can undermine interpretation. The standard joint-model formulation also requires a defensible account of how visits and measurements arise; if visit timing depends on worsening health, treating the schedule as noninformative may be inadequate.
Worked Example: Interpreting a Current-Value Association
Worked Example: Interpreting a Current-Value Association
Suppose a hypothetical joint-model fit gives \(\hat{\alpha}=0.55\), with standard error \(0.20\), for latent log CA-125. Assume for this calculation that the model’s current-value association and proportional hazards form are appropriate. What is the hazard ratio for a doubling of the latent CA-125 concentration, and what does a Wald interval indicate?
A doubling is a change of \(\log(2)\) in log concentration. The estimated hazard ratio is
Thus, under the fitted model, a doubling in latent CA-125 is associated with an estimated 1.464-fold hazard of the defined recurrence event, conditional on the model terms. The unit is a hazard ratio, not a difference in probability or a claim about causation.
A 95% Wald interval for \(\alpha\) is \(0.55\pm1.96(0.20)\), or \(0.158\) to \(0.942\). Transform the endpoints for a doubling:
Under the model and its sampling assumptions, this interval procedure has 95% coverage in repeated samples. Here the interval is compatible with hazards from about 1.116 to 1.921 times as high per doubling. The computation checks the scale of the interpretation; it does not verify that the event definition, visit process, or model form is suitable.
Surveillance Timing and What an Event Date Means
Suppose visits occur at months 12 and 15, with CA-125 below a prespecified threshold at month 12 and above it at month 15. The observed measurements bracket a change between those two observations. They do not establish the exact crossing time. Nor, without additional assumptions, do they establish the first crossing time: the biomarker could have crossed earlier and then fallen below the threshold before month 12.
If threshold-event status is assumed to be absorbing (once reached, it remains reached) and the visits correctly classify that status, the first threshold event is known to have occurred after month 12 and by month 15. Those are substantive assumptions, not consequences of the two measurements. If they are not defensible, describe only the observed results and the bracketed change; do not label the interval as the first crossing time.
Worked Example: A Threshold Rule and an Interval of Uncertainty
In a hypothetical surveillance protocol, a CA-125 threshold of 35 U/mL defines a biomarker event when the threshold is first met. A patient has a value of 31 U/mL at month 12 and 42 U/mL at month 15. Assume threshold status is absorbing and both measurements correctly classify status. What can be said about the event time?
At month 12 the patient is below threshold; at month 15 the patient is above it. Under the stated assumptions, the first threshold event time lies in \((12,15]\) months. The data do not identify a specific month within that interval. If the analysis instead records the first observed above-threshold visit, the operational event date is month 15, which is a detection time under this visit schedule rather than a known biological onset time.
If the absorbing-status assumption is removed, the two values still show that a crossing between these observed values is bracketed, but they do not establish when the first crossing occurred. A prior crossing followed by a decline is possible. This distinction matters when interpreting event times or comparing patients with different visit schedules.
Clinical recurrence may be defined by imaging, pathology, clinician adjudication, symptoms, a biomarker rule, or a specified combination. A date based on confirmed disease is a detection or confirmation date; it need not equal biological onset. If CA-125 itself is part of the event rule, the association between CA-125 and that event partly reflects how the outcome was constructed. This is not automatically invalid, but it changes the scientific question: the result concerns the defined biomarker-based endpoint, not an independent measure of biological recurrence.
Worked Example: How the Recurrence Definition Changes the Analysis
Consider a hypothetical cohort of 80 patients followed from treatment completion. Under a clinical endpoint requiring imaging-confirmed recurrence, 12 patients have events during 400 total person-years of observed follow-up. Under a separate biomarker-only rule, 8 additional patients meet the threshold rule before any imaging confirmation. Assume, for this illustration, these 8 are not among the 12 clinical events during the stated follow-up.
For the clinical endpoint, the crude event rate is \(12/400=0.0300\) events per person-year, or 3.00 per 100 person-years. If the biomarker-only cases are instead counted as events and their observed follow-up contributes 40 additional person-years, the combined endpoint has \(20/440=0.0455\) events per person-year, or 4.55 per 100 person-years. These crude rates are not joint-model estimates; they simply show how changing the event definition changes the event count and denominator.
The second endpoint asks about first occurrence of either clinical confirmation or the specified biomarker rule, not imaging-confirmed recurrence alone. A joint-model coefficient fitted to that endpoint must be interpreted accordingly. Because the biomarker contributes directly to classifying some events, an association with CA-125 can be strengthened by endpoint construction as well as by any underlying relation to disease.
Diagnostics, Alternatives, and Limitations
A joint model can help when the longitudinal marker and event process are associated and follow-up measurements are incomplete because an event or censoring ends observation. It does not make surveillance design irrelevant. Inspect measurement times and their relation to observed clinical status: unusually frequent testing near suspected recurrence may signal a visit process related to disease. If visit intensity is informative and not represented in the model, trajectory estimates and event associations may be affected.
Check whether the fitted longitudinal trajectories track the observed log CA-125 values across time and across patients; look for systematic residual patterns, influential patients, implausible extrapolation, and sensitivity to the time trend or transformation. For the event component, examine whether the assumed baseline hazard and covariate specification are credible, and whether there are enough events to support the parameters. Numerical convergence and a finite standard error are necessary computational checks, not proof that the endpoint or assumptions are scientifically appropriate.
Separate longitudinal and time-to-event analyses can be simpler, but they may not represent the dependence between evolving CA-125 and event risk, especially when measurements stop as event-free follow-up ends. A landmark analysis can instead relate information available at a chosen time to subsequent risk among patients still at risk; its interpretation is conditional on reaching that landmark and it does not answer exactly the same question as a full joint model. The choice depends on the estimand and design, not only on which method produces a more favorable result.
No model can recover an exact biological recurrence time from sparse surveillance data without additional information or assumptions. A current-value association also does not imply that the biomarker is a causal driver, and a well-fitting model cannot repair a poorly defined endpoint. Report the time origin, visit schedule, threshold and confirmation rules, event-date convention, censoring approach, and model association scale so that the result can be interpreted in context.
A Practical Interpretation Sequence
State whether recurrence means clinical confirmation, a biomarker rule, or a composite, and say how its date is assigned.
Give the CA-125 scale, visit schedule, time origin, and any threshold or confirmation procedures.
For a log current-value effect, translate \(\alpha\) into a hazard ratio for a stated change, such as a doubling.
Relate the estimate to the defined endpoint and model assumptions; distinguish detection from biological onset and association from causation.
Check Your Understanding
Answer each question using the stated model and endpoint definitions. Short solutions follow.
- A fitted current-value coefficient is \(\hat{\alpha}=0.40\) for log CA-125. What is the estimated hazard ratio for a doubling?
- A patient is below threshold at month 6 and above it at month 9. What additional assumptions allow the first threshold-event time to be placed in \((6,9]\)?
- Why can a CA-125 association appear stronger when CA-125 is included in the recurrence definition?
- What does an event date based on the first visit with imaging-confirmed recurrence represent, and what does it not necessarily represent?
- Give one reason that an ordinary joint model may be inadequate when surveillance visits become more frequent as a patient worsens.
Solutions. (1) A doubling changes log CA-125 by \(\log(2)\), so the hazard ratio is \(\exp\{0.40\log(2)\}=\exp(0.2773)\approx1.320\). (2) Threshold status must be absorbing, and visits must correctly classify that status; otherwise an earlier crossing followed by a decline cannot be ruled out. (3) The biomarker helps determine which observations count as events, so the association partly reflects endpoint construction and is not solely evidence of an independent biological relationship. (4) It is a confirmation or detection time under the surveillance process; biological onset may have occurred earlier. (5) Visit timing may carry information about disease status. If that informative observation process is ignored, the assumed measurement process may be inappropriate and inference about trajectories or event association may be biased.