Accrual Rate and Observation Windows in TITE-CRM

Why Timing Matters in TITE-CRM

In a time-to-event continual reassessment method (TITE-CRM), an update can occur before every enrolled patient has completed the toxicity-observation window. The timing of enrollment therefore determines how many outcomes are resolved and how much partial follow-up is available when a dose decision is made. Faster accrual or a longer observation window will generally leave more patients with unresolved outcomes at an update; slower accrual or a shorter window will generally leave fewer.

That relationship is operationally important, but it is not a toxicity result by itself. A large pending group does not show that a dose is safe or unsafe. Its contribution to the dose decision depends on the patients’ assigned doses, observed follow-up, toxicity events, the TITE-CRM likelihood and its chosen follow-up weights, and the decision rule.

Definition: Let patient \(i\) enter the trial at time \(a_i\), and let \(t\) be an update time. If the toxicity-observation window has length \(W\), that patient’s follow-up age at the update is \(u_i=t-a_i\). A patient is pending when \(0\leq u_i<W\) and no dose-limiting toxicity (DLT) has yet been observed. A patient with an observed DLT is not pending, even if the window is not complete. A patient with \(u_i\geq W\) and no DLT has completed the window without an observed DLT.

This definition assumes that the observation window starts at enrollment, that toxicity status and follow-up time are recorded accurately, and that the same prespecified window is used for the patients being counted. If the protocol defines a different time origin or window, use that definition instead. A count based only on enrollment dates is a count of patients whose windows are still open; it is not necessarily the count of pending outcomes, because some patients may already have had a DLT.

Counting Patients With Open Windows

For a fixed update time, count all patients who entered after the window’s start boundary and no later than the update. If \(N(t)\) is the cumulative number enrolled by time \(t\), the number whose observation windows have not yet elapsed is

M_{\text{open}}(t)=N(t)-N(t-W).

Here the count includes an enrollment at \(t\), whose follow-up age is zero, and excludes an enrollment at \(t-W\), whose window has just elapsed. To obtain the number actually pending, remove from this count any patients with an observed DLT. The formula assumes enrollment times and update times use a consistent convention for boundary cases; a protocol’s operational rules determine how to handle observations occurring exactly at a boundary.

Worked Example: Counting Open Windows at a Calendar-Time Update

Suppose, hypothetically, that patients enroll every 14 days, at days 0, 14, 28, 42, 56, and 70. The toxicity window is 42 days, and the update occurs immediately after the day-70 enrollment. Assume no DLT has been observed.

The window began at day \(70-42=28\). Enrollments after day 28 and no later than day 70 are on days 42, 56, and 70. The day-28 patient has completed 42 days and is therefore not pending.

Thus \(M_{\text{open}}(70)=N(70)-N(28)=6-3=3\). All three open-window patients are pending under the stated assumption of no observed DLT. If the day-56 patient had already experienced a DLT, there would still be three open windows but only two pending outcomes.

A constant accrual rate gives a useful planning approximation. Let \(\lambda\) be the average enrollment rate in patients per unit of time, and suppose accrual has been running for at least \(W\) time units. The expected number enrolled in the most recent \(W\) time units is approximately \(\lambda W\). Before accrual has run for \(W\) time units, replace \(W\) with the elapsed accrual duration. This approximation describes open windows; the number pending can be smaller when DLTs have already been observed.

Planning approximation: With roughly constant accrual and an update during steady accrual, expected open-window occupancy is approximately \(\lambda W\). Doubling the accrual rate or doubling the window length approximately doubles this occupancy, if the other quantity stays fixed.

For example, an accrual rate of 1 patient per week and a 6-week window suggest roughly 6 patients enrolled in the most recent window-length of time. This does not mean exactly 6 patients will be pending at every update. Actual enrollment may vary, some patients may have observed DLTs, and update timing may not coincide with a regular weekly schedule.

Variable Accrual and the Pending-Count Distribution

When enrollment is irregular, a single average rate can conceal substantial variation. One standard planning model assumes arrivals follow a homogeneous Poisson process: counts in disjoint time intervals are independent, and the expected number enrolled in an interval is its length multiplied by \(\lambda\). Under this model, once accrual has been running for at least \(W\), the number of arrivals in a window of length \(W\) is Poisson with mean \(\lambda W\). The model is an approximation, not a requirement of TITE-CRM; it is most useful when enrollment is reasonably stable and individual arrivals are not strongly coordinated.

P(M_{\text{open}}=m)= \frac{e^{-\lambda W}(\lambda W)^m}{m!}, \qquad E(M_{\text{open}})=\lambda W, \qquad \operatorname{Var}(M_{\text{open}})=\lambda W.

The expectation follows by summing over the Poisson probabilities:

E(M_{\text{open}}) =\sum_{m=0}^{\infty}m\frac{e^{-\lambda W}(\lambda W)^m}{m!} =\lambda W e^{-\lambda W}\sum_{r=0}^{\infty}\frac{(\lambda W)^r}{r!} =\lambda W.

The last equality uses the exponential series, \(\sum_{r=0}^{\infty}x^r/r!=e^x\). This derivation concerns enrollment into open windows, not the number of pending patients after accounting for DLTs. If enrollment rates change over calendar time, or if accrual occurs in planned cohorts, use the actual schedule or a more suitable arrival model rather than assuming a homogeneous Poisson process.

Worked Example: Planning for Variation in Pending Counts

Suppose, hypothetically, a trial is expected to enroll at an average rate of 1.5 patients per week, with a 4-week toxicity window. Assume steady homogeneous Poisson accrual and consider open windows before subtracting patients with observed DLTs.

The mean count is \(\lambda W=1.5\times4=6\) patients. The Poisson variance is 6, so the standard deviation is \(\sqrt{6}=2.45\) patients. A mean of 6 is a planning average, not a promise that each update will have exactly 6 open windows.

If the window were shortened to 2 weeks at the same rate, the mean would be \(1.5\times2=3\) patients and the standard deviation would be \(\sqrt{3}=1.73\). If instead the rate rose to 2 patients per week while the 4-week window remained, the mean would be \(2\times4=8\). These comparisons show how both design choices affect the expected information lag, without asserting anything about the underlying toxicity probabilities.

From Pending Counts to Dose Decisions

TITE-CRM uses follow-up information from patients whose toxicity windows are incomplete according to a prespecified likelihood and follow-up-weight scheme. The earlier material on choosing follow-up weights explains the choice of those weights; the key timing point here is that the patient’s follow-up age, measured against \(W\), determines how much observation has accrued at an update. A pending patient with only a short time under observation does not contribute the same information as a patient with longer follow-up. A patient with an observed DLT contributes event information rather than remaining in the pending group.

Consequently, accrual rate and window length can change the data available at a decision time, even when the dose assignments and eventual outcomes would otherwise be the same. More open windows usually mean more outcomes still unresolved and more dependence on partial follow-up. Fewer open windows usually mean that a larger share of enrolled patients has reached a complete toxicity assessment. The direction of the resulting dose change cannot be inferred from the pending count alone: it depends on which doses the pending patients received and how their follow-up contributes under the fitted model.

Worked Example: Pending Follow-Up Can Change Eligibility

Consider a hypothetical, deliberately small grid approximation to a monotone dose-toxicity model with two possible toxicity profiles, A and B, each assigned prior probability 0.5. At dose 1, the DLT probabilities are 0.10 under A and 0.20 under B. At dose 2, they are 0.25 under A and 0.50 under B. Suppose the target toxicity probability is 0.30, and a dose is eligible only if its posterior probability of exceeding 0.30 is at most 0.25. This simplified calculation illustrates how partial follow-up can affect a dose rule; it is not a substitute for fitting the trial’s specified CRM model.

Assume six patients at dose 2 are pending, each with follow-up weight \(w=0.5\), and that the specified likelihood gives a no-DLT pending patient a contribution \((1-p)^w\) when the dose-specific DLT probability is \(p\). Under profile A, the combined likelihood contribution is

L_A=(1-0.25)^{6(0.5)}=0.75^3=0.421875.

Under profile B, it is

L_B=(1-0.50)^{6(0.5)}=0.50^3=0.125.

With equal prior probabilities, the posterior probability of profile B is

P(B\mid\text{follow-up}) =\frac{0.5L_B}{0.5L_A+0.5L_B} =\frac{0.125}{0.421875+0.125} =0.2286.

Only profile B exceeds the 0.30 target at dose 2, so the posterior probability that dose 2 exceeds the target is 0.2286. This is below the hypothetical eligibility threshold of 0.25; dose 2 is eligible. Dose 1 is also eligible, because neither profile’s dose-1 probability exceeds 0.30, so the rule selects the higher eligible dose, dose 2. With no follow-up information, the posterior probability of profile B would still be 0.5, making dose 2 ineligible and leaving dose 1 as the highest eligible dose. The example shows how the information accumulated by an update can alter eligibility; actual decisions require the prespecified CRM model, likelihood, weights, and decision rule.

Operational Checks and Alternative Update Strategies

The count calculation is simple, but its interpretation depends on reliable timing and outcome records. At each update, check enrollment dates, each patient’s follow-up age, the protocol-defined window, observed DLT dates, and whether follow-up is ongoing or complete. Compare the observed open-window count with the planning expectation \(\lambda W\), while remembering that a mismatch may reflect variable accrual rather than a modeling error. Also inspect the distribution of follow-up ages: a count of six patients all near the beginning of their windows represents a different amount of observed follow-up from six patients near completion.

A fixed-cohort approach that waits for every patient’s full observation window before updating avoids making decisions with pending outcomes, but can delay decisions and enrollment. TITE-CRM permits updates during follow-up, potentially reducing that delay, but its decisions then depend on the model’s treatment of incomplete observation. Excluding all pending patients is another possible analysis choice, but it discards partial follow-up and can make the available data depend strongly on how quickly patients were enrolled. These approaches trade timeliness against the amount and type of information used; no pending-count formula alone determines which is appropriate.

Window definition: Changing \(W\) changes which patients have completed follow-up and the amount of partial follow-up represented at an update.
Accrual pattern: Bursts or pauses can make actual open-window counts depart from a constant-rate approximation.
Observed DLTs: Patients with an event are resolved for the DLT endpoint, even if their planned window has not elapsed.
Interpretation: More pending patients do not automatically justify escalation or de-escalation. Their doses, follow-up ages, and model-based contributions matter.

Key takeaway: Accrual rate and observation-window length determine how much follow-up can be available at a calendar-time update. Use actual dates to count open windows and pending outcomes, then let the prespecified TITE-CRM likelihood and dose rule—not the count by itself—determine the dose decision.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Patients enroll on days 0, 10, 20, 30, and 40. At a day-40 update, the observation window is 30 days. Assuming no DLTs, how many patients have open windows?
At a constant accrual rate of 2 patients per week, what is the approximate mean number of open windows for a 5-week observation window after steady accrual begins?
Why can the number of patients enrolled in the last \(W\) time units differ from the number of pending outcomes?
Does a larger pending count by itself imply that the next dose should be higher or lower? Explain.

Solutions. (1) The window began on day \(40-30=10\). Enrollments after day 10 and through day 40 occurred on days 20, 30, and 40, so there are 3 open windows. (2) The approximate mean is \(\lambda W=2\times5=10\) patients. (3) Some patients enrolled within the window may already have had an observed DLT; they have an open window by calendar time but are not pending. (4) No. The dose decision depends on dose assignments, follow-up and observed outcomes, the fitted TITE-CRM model, and the prespecified decision rule—not the pending count alone.