Why Predose BQL Records Need Special Attention
A below-quantification-limit (BQL) result collected before a dose can look like evidence that the drug concentration is zero. But the record alone does not establish that conclusion. It may represent an assay of a collected sample whose result fell below the assay’s lower limit of quantification (LLOQ); it may arise when the model prescribes a structural zero; or it may be a placeholder for a sample that was never collected or had no usable result. These cases have different meanings and should not be encoded interchangeably.
The key comparison is between the assay threshold and the concentration predicted by the model immediately before dosing. A BQL assay result can be informative when that predicted baseline is not negligible relative to the threshold. Conversely, if the model fixes baseline concentration at zero because the subject had no prior exposure, a BQL result may add little or no information about the pharmacokinetic parameters. Correct classification therefore requires both the assay record and the dosing and sampling history.
Modeling the Predose Event
Let \(C_{i0}\) denote the model-predicted concentration for subject \(i\) at the predose time, conditional on the population parameters and that subject’s random effects. Write this dependence as \(C_{i0}=C_i(0^-;\theta,\eta_i)\), where \(0^-\) means immediately before the dose and \(\theta\) and \(\eta_i\) denote the fixed effects and individual random effects. The value \(C_{i0}\) must reflect the model’s dosing history: it is not automatically zero simply because a sample is labeled “predose.”
Suppose the assay-scale observation has a conditional density \(f(y\mid C_{i0},\psi)\) and cumulative distribution function \(F(y\mid C_{i0},\psi)\), where \(\psi\) denotes residual-error parameters. The model must specify the distribution and the scale on which it applies. A collected sample reported BQL contributes the probability of the event below the threshold, conditional on the individual’s model values:
This is the predose version of a left-censored likelihood contribution. It is not the likelihood of observing an exact value \(L_i\), nor is it a statement that the underlying concentration equals zero. In a hierarchical model, the contribution is conditional on \(\eta_i\); the subject-level likelihood is then handled using the model’s usual integration over random effects.
For illustration, under additive normal error on the assay scale, \(Y_{i0}\mid C_{i0}\sim N(C_{i0},\sigma^2)\). Then
where \(\Phi\) is the standard normal cumulative distribution function. This formula is an example, not a universal residual model. The observation distribution must be appropriate for the assay scale and the model being fitted.
Worked Example: A Predose BQL Result With Nonzero Baseline
Suppose a hypothetical predose sample was collected and reported BQL, with \(L=0.050\) mg/L. At that time, the fitted model predicts a baseline concentration \(C_0=0.080\) mg/L, and the illustrative additive-error model has \(\sigma=0.020\) mg/L. Calculate the conditional likelihood contribution for the BQL event.
The sample is observed, so the event is \(Y_0<0.050\) mg/L. Under the stated normal observation model,
Thus, conditional on this model prediction and error scale, the BQL record contributes about 0.0668 to the likelihood. This is relatively small: a baseline prediction above the threshold makes a BQL assay result less expected, though not impossible under the assumed error model. It does not justify replacing the unobserved value with zero.
Three Different Data States
A predose record should be assigned to the data state supported by the study records and the model. A BQL report is an observed assay outcome; a structural zero is a model or design statement about the underlying concentration; missing data indicate that no usable assay outcome is available. The distinction affects both the likelihood and the interpretation.
| Situation | What is known | How it enters the analysis |
|---|---|---|
| Collected sample reported BQL | The assay result was below the applicable LLOQ. | Use the probability of the below-threshold event under the observation model. |
| Structural zero | The model or design fixes the relevant underlying concentration at zero. | Represent the constraint in the model. Do not fabricate an assay result; include an assay-event likelihood only if a sample was actually measured. |
| Missing predose sample or result | No usable measurement or BQL event is observed. | There is no assay-value likelihood contribution for that time point. |
A structural zero and a BQL assay result can occur together, but they are not synonyms. For example, a subject with no prior exposure may have a structurally zero predose drug concentration, and a collected assay may also be reported BQL. The structural zero describes the model state; the BQL record describes the assay outcome. Under an observation model with error, the event probability at zero is generally not exactly one. Whether it informs parameters depends on the model: if the predose mean is fixed at zero independently of the pharmacokinetic parameters, its contribution may be constant with respect to those parameters, although it could still depend on error parameters.
If a study’s convention uses “zero” only as a data-entry code for “not detected,” that code must not be treated as a measured concentration without verification. Likewise, a blank cell, an explicit “not collected” flag, and an assay result marked BQL are not interchangeable records.
Worked Example: Structural Zero With an Observed BQL Assay
Suppose a hypothetical subject has no prior exposure, and the model therefore fixes the predose concentration at \(C_0=0\) mg/L. A sample was nevertheless collected, the LLOQ was \(L=0.050\) mg/L, and the additive normal error standard deviation is \(\sigma=0.020\) mg/L. Calculate the BQL-event probability, and distinguish it from the structural-zero statement.
Under the stated assay observation model,
The collected assay result contributes approximately 0.9938 for the BQL event under this illustrative model. Separately, the model’s structural-zero assumption says that the predose concentration is fixed at zero. Since \(C_0\) is fixed rather than estimated through the pharmacokinetic parameters in this example, this event probability does not distinguish values of those parameters. It is not correct to claim that a BQL record itself proves the structural zero.
If instead “structural zero” means the data contain no assay event and the model imposes a zero state by design, there is no BQL likelihood factor to calculate. The likelihood contribution above is appropriate only because the example specifies a collected sample with a reported BQL result and an observation model.
Worked Example: Missing Predose Result Is Not a BQL Event
In a hypothetical dataset, a predose sample was scheduled for a subject with a model-predicted baseline of \(0.030\) mg/L and an LLOQ of \(0.050\) mg/L. The collection log says the sample was not obtained, and the assay field is blank. Should this record contribute the BQL probability under the same illustrative model with \(\sigma=0.020\) mg/L?
No. The collection log establishes that there is no assay result and no observed event \(Y_0<0.050\). The value
would be the likelihood contribution if a collected sample had actually been reported BQL. It is not the contribution for a sample that was not obtained. Under the usual analysis of observed assay outcomes, the missing time point supplies no assay-value likelihood factor. Any assumptions used to handle missingness belong to the analysis plan; they do not turn this blank record into a censored observation.
Classification and Model Checks
The following sequence helps prevent a predose label from determining the analysis by itself.
Check collection, assay, and data-entry records. Establish whether the result is explicitly BQL, quantified, missing, or only represented by a code whose meaning is uncertain.
Use the LLOQ applicable to that sample, including any assay or batch-specific threshold, and verify that “predose” is relative to the correct dose time.
Determine whether the model predicts a baseline concentration from prior dosing or fixes it at zero. Do not infer the baseline solely from the BQL label.
For a genuine BQL assay result, use the model probability below the threshold. For a quantified result, use its density; for a missing sample, do not invent an assay event.
A useful basic check is whether a BQL record is plausible under the predicted baseline and residual-error distribution. If \(F(L_i\mid C_{i0},\psi)\) is extremely small for many predose BQL samples, inspect the baseline model, the dosing history, threshold units, sample times, and assay coding. A very small probability is a model-data conflict to investigate, not evidence that the record should be recoded as zero. Similarly, a quantified predose result should be compared with its predicted distribution and the applicable LLOQ.
Predose records can also expose a mismatch in the model’s assumptions. A model that fixes every predose concentration to zero may be unsuitable if subjects could have residual drug from earlier exposure. Conversely, fitting a free baseline concentration when the study design and verified history establish no prior exposure may add an unsupported parameter. The decision should follow the scientific and data-record context, not whichever choice makes the likelihood more convenient.
Alternatives, Pitfalls, and Limits
For a genuine collected BQL sample, the censored-likelihood approach retains the information that the result lies below the threshold without pretending to know its exact value. Deleting the record discards that event information. Substituting a number such as half the LLOQ assigns an exact value the assay did not provide and changes the likelihood as though that value had been measured. Neither approach repairs a mistaken classification of a missing sample as BQL or of a structural zero as an assay result.
The censored likelihood is only as credible as its assumptions. It requires the correct threshold, observation distribution, baseline prediction, and sampling history. If the assay result is mislabeled, the observation model is inappropriate, or the model’s predose concentration is inconsistent with prior exposure, the calculated probability may be mathematically correct for the wrong setup. A single plausible BQL probability also does not establish that the entire model is adequate.
Correction: First establish whether a sample was collected and reported below the applicable LLOQ. If it was, the observed event is below-threshold and its probability is evaluated using the model-predicted baseline and observation distribution. A structural zero is a separate model statement, and an unavailable sample is missing—not BQL.
Check Your Understanding
Answer each question by identifying the data state and, where appropriate, the likelihood contribution.
- A collected predose sample is reported BQL at \(L=0.040\) mg/L. What event does the record represent?
- Under additive normal error, \(C_0=0.060\) mg/L, \(L=0.040\) mg/L, and \(\sigma=0.010\) mg/L. What is the conditional BQL probability?
- A subject’s predose concentration is fixed at zero by verified study design, but no sample was collected. Is there a BQL assay likelihood contribution?
- Why is replacing a predose BQL value with zero not generally equivalent to using a censored likelihood?
Solutions. (1) It represents the observed event \(Y_0<0.040\) mg/L, not an exact concentration of zero. (2) The probability is \(\Phi((0.040-0.060)/0.010)=\Phi(-2.00)\approx0.0228\). (3) No. The structural-zero condition may be part of the model, but without an observed assay result there is no BQL event likelihood. (4) Zero is an invented exact value, whereas the censored likelihood uses the probability of any assay result below the threshold under the specified observation model.