Why the Position of a BQL Run Matters
A sequence of below-quantification-limit (BQL) records is not a sequence of measured concentrations equal to zero or to the assay limit. It records that the observed concentrations fell below specified limits. In the terminal phase, when a concentration profile is expected to decline, consecutive BQL records can be entirely consistent with the model: a profile may simply have crossed the assay’s quantification limit before the last scheduled samples.
The timing of the run matters. A run beginning well after the predicted profile has fallen below the limit is usually unsurprising. A run beginning while the model predicts concentrations clearly above the limit can be evidence worth investigating. It is not, by itself, proof that the elimination model is wrong: residual error, assay variation, sampling, and individual deviations also affect the probability of BQL observations.
A useful interpretation therefore has two parts: first, ask what the proposed terminal profile predicts at each time; second, assess how likely the observed BQL events are under the observation model. The earlier material on left censoring and censored likelihoods establishes the event-based treatment of BQL records. Here the emphasis is on what a consecutive pattern says about a declining terminal profile.
Terminal Profile and Censoring Assumptions
A common local approximation in the terminal phase is a monoexponential decline. For a particular subject, write the model-predicted concentration at time \(t\) as
where \(t_a\) is a time in the terminal phase, \(C(t_a)>0\) is the predicted concentration at that time, and \(\lambda_z>0\) is the terminal elimination rate constant, in inverse time units. This approximation assumes that absorption is complete, the terminal decline is adequately represented by one exponential over the times considered, and the model parameters are fixed for the subject in the calculation. It is a local description, not a claim that every PK profile is monoexponential throughout.
For a lognormal residual model, suppose
with independent residual errors across observations, conditional on the subject-specific model parameters. Here \(\sigma\) is the standard deviation on the log scale. If the observation is BQL at limit \(L_j\), its conditional probability under this model is
where \(\Phi\) is the standard normal cumulative distribution function. This expression compares the log LLOQ with the log predicted concentration in units of residual standard deviations. A probability near one means a BQL observation is expected under the specified model; a probability near zero means it is unusual. If a different residual distribution is used, the corresponding probability must be calculated from that distribution instead.
For consecutive records at times \(t_1,\ldots,t_m\), the probability that all are BQL is the product of their conditional probabilities when residual errors are independent and the subject-specific profile is held fixed:
This is a conditional statement. In a population analysis, subject-specific random effects are not generally known; inference must account for them rather than treating a plug-in subject profile as certain. Also, the product formula does not apply if residual errors are dependent unless that dependence is included in the joint probability model.
What the Run Says About a Declining Curve
A simple way to understand the information in a run is to temporarily ignore residual error and treat the predicted concentration as the value being compared with the LLOQ. Then a BQL record at time \(t_j\) implies \(C(t_j)<L_j\). For a common LLOQ \(L\), an anchor concentration \(C(t_a)>L\), and a later BQL time \(t_j>t_a\), substituting the terminal profile gives
Taking logarithms and rearranging yields
Thus, under this deterministic simplification, a BQL observation places a lower bound on how quickly the profile must decline from the anchor. With the same LLOQ at later and later times, these lower bounds become weaker: a concentration that is below the limit at an earlier time will remain below it under a strictly declining profile. Consequently, a long run is not automatically many independent pieces of strong slope information. The earliest BQL time after a quantified anchor may impose the most restrictive bound.
With residual error, BQL does not impose a hard bound on the model-predicted concentration. A prediction above the LLOQ can still produce a BQL result, and a prediction below it can produce a quantified result. The inequality is therefore a useful geometric diagnostic, not a substitute for the censored observation likelihood.
Worked Example: Probability of an Early BQL Run
Suppose a hypothetical subject has a predicted terminal profile \(C(t)=1.20\exp(-0.100t)\) mg/L, where \(t\) is in hours. Assume a common LLOQ of \(0.500\) mg/L and log-scale residual standard deviation \(\sigma=0.200\). Calculate the conditional probability of BQL at 4, 8, and 12 hours, and the probability that all three records are BQL.
At 4 hours, \(C(4)=1.20\exp(-0.100\times4)=1.20\exp(-0.4)=0.8044\) mg/L. The standardized log threshold is
Therefore, \(P(Y_4<0.500)\approx\Phi(-2.377)=0.0087\). Under this profile, a BQL record at 4 hours is uncommon.
At 8 hours, \(C(8)=1.20\exp(-0.8)=0.5392\) mg/L, so
At 12 hours, \(C(12)=1.20\exp(-1.2)=0.3614\) mg/L, giving
With independent residual errors conditional on this profile, the probability of all three BQL records is approximately \(0.0087\times0.353\times0.9477=0.00291\), or 0.291%. The run beginning at 4 hours is unusual under the stated assumptions, largely because the model predicts a concentration well above the LLOQ then. That result motivates review of the profile, residual model, and assay context; it does not establish which assumption, if any, failed.
Worked Example: A Run That Is Expected Late in the Terminal Phase
Consider the same hypothetical profile, LLOQ, and residual standard deviation as above, but suppose the subject’s records are quantified at 4 and 8 hours and BQL at 12 and 16 hours. Determine whether the two-record BQL run is compatible with the profile.
The predicted concentrations are \(C(12)=1.20\exp(-1.2)=0.3614\) mg/L and \(C(16)=1.20\exp(-1.6)=0.2423\) mg/L. At 12 hours, the BQL probability is approximately \(0.9477\), as calculated above. At 16 hours,
The conditional probability of both BQL records is approximately \(0.9477\times0.9999=0.9476\). The run is therefore expected under this profile. The quantified records at 4 and 8 hours do not contradict that interpretation: their predicted concentrations, \(0.8044\) and \(0.5392\) mg/L, are above the LLOQ, and quantified results at those times are plausible.
This calculation depends on the assumed profile and residual model. It is not a universal probability for a late run: different LLOQs, residual variability, sampling times, or subject-specific parameters can change the result.
Worked Example: Using the First BQL Time as a Slope Diagnostic
In a separate hypothetical case, suppose a terminal-phase anchor prediction is \(C(4)=0.800\) mg/L, the LLOQ is \(0.500\) mg/L, and observations at 8 and 12 hours are BQL. First use the deterministic threshold calculation to see what slope is required for the prediction to be below the limit.
At 8 hours, the elapsed time from the anchor is 4 hours. The required rate is
At 12 hours, the elapsed time is 8 hours, so the corresponding bound is
The earlier BQL time gives the stricter bound. A proposed rate of \(0.100\ {\rm h}^{-1}\) falls below the deterministic 8-hour bound. Yet with log-scale \(\sigma=0.200\), the predicted concentrations are \(0.800\exp(-0.4)=0.5363\) mg/L at 8 hours and \(0.800\exp(-0.8)=0.3595\) mg/L at 12 hours. The corresponding BQL probabilities are approximately
Their product is approximately \(0.345\), conditional on the stated profile and independent errors. Thus, the run is possible under the stochastic observation model even though the deterministic bound flags the first BQL time as early for the proposed slope. The two calculations answer different questions; the likelihood-based probability is the relevant one when residual error is part of the model.
Diagnostics, Alternatives, and Limitations
When a run appears surprising, evaluate the whole context rather than changing the elimination rate solely to accommodate the BQL records. A structured review can help separate a genuine profile problem from a plausible censoring pattern.
Check the sampling times, units, assay assignment, and LLOQ attached to each observation. A mistaken limit or time can make an otherwise plausible run appear anomalous.
Check that the samples are after absorption and that the proposed interval is reasonably described by a declining terminal profile. A decline need not be monoexponential across the entire concentration-time course.
At each time, calculate the predicted concentration and the probability of falling below that record’s LLOQ under the chosen residual model. Pay particular attention to the first BQL time after a quantified terminal observation.
Consider whether the run reflects residual variability, an individual profile not represented by the fitted parameters, an unsuitable terminal slope, or an assay or data-handling issue. A single run usually cannot identify which explanation is correct.
The censored-likelihood approach uses the BQL event itself and retains the information that the observation lies below its limit. In contrast, deleting BQL records discards that information. Substituting zero, the LLOQ, or a fraction of the LLOQ assigns a value that was not observed and can distort the apparent terminal slope. Deterministic threshold reasoning is useful for seeing how a run constrains a curve, but it ignores residual error and should not be mistaken for a full stochastic analysis.
Several limitations remain even when the calculations are correct. The probability in a worked calculation is conditional on the specified profile and residual parameters, which are themselves estimated with uncertainty. A monoexponential terminal approximation can be inadequate, and independent residual errors may not describe the data. Population heterogeneity can also make a single fixed-profile calculation unrepresentative. Consequently, a low conditional probability is a diagnostic signal to investigate, not a formal proof of model failure or a causal explanation for the observations.
Common Pitfall
Correction: Each BQL record means only that its observed concentration was below its applicable LLOQ. Under a declining profile, a late run may be very likely. An early run can be unusual under a proposed model, but its interpretation depends on the predicted concentrations, residual variability, limits, sampling times, and model assumptions. Do not replace BQL values with zero or treat the number of records alone as evidence against the profile.
The central diagnostic is therefore not simply “How many BQL records occur?” It is “At what times do they occur relative to the predicted decline and the relevant LLOQs, and how probable is that pattern under the observation model?” That framing preserves the distinction between censoring information and evidence that the structural profile may need revision.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A terminal model predicts concentrations below the LLOQ at all three times in a consecutive BQL run. Does the run, by itself, imply a poor elimination profile?
- With a fixed anchor concentration above a common LLOQ and no residual error, which gives the stricter lower bound on \(\lambda_z\): an earlier BQL time or a later BQL time?
- Why is multiplying the conditional probabilities of consecutive BQL observations justified only under particular assumptions?
- A BQL record occurs when the model predicts a concentration above the LLOQ. Is that logically impossible under a lognormal residual model?
Solutions. (1) No. If the predicted concentrations are below the limits, the run can be expected; its probability still depends on the observation model. (2) The earlier BQL time gives the stricter lower bound because the numerator is the same while the elapsed time in the denominator is smaller. (3) The product requires independent residual errors conditional on the fixed subject-specific parameters; dependence or uncertainty in shared subject effects requires a joint or appropriately integrated calculation. (4) No. A positive residual error model allows an observed concentration to fall below the LLOQ even when the predicted concentration is above it; the probability may be small or substantial depending on the separation and residual variability.