BQL Records Before the First Quantified Sample

Why Early BQL Records Matter

After an oral dose, drug concentration may rise as absorption proceeds. A sample collected before the first quantified concentration can therefore be below the assay’s lower limit of quantification (LLOQ) even when absorption has begun. Such a record does not say that the concentration was zero. It says that the assay result was below a known threshold at a known time. Together with later quantified samples, early BQL records inform whether the model places exposure too early, too high, or plausibly below the assay limit.

The key comparison is between each observed BQL status and the model-predicted concentration at that sample time. A BQL record is more compatible with predictions well below the LLOQ than with predictions far above it, but residual variability means compatibility is probabilistic, not deterministic. Early BQL records can constrain the predicted rise in concentration without uniquely identifying an absorption parameter.

Likelihood Contribution and Its Meaning

For a transparent calculation, suppose the assay result on the original concentration scale follows an additive normal residual model conditional on the model-predicted concentration \(\mu\):

Here \(\sigma\) is the residual standard deviation, in concentration units. A quantified result contributes a density evaluated at its measured value. A BQL result contributes the probability that \(Y\) falls below \(L\). With \(\Phi\) denoting the standard normal cumulative distribution function,

This expression is specific to the stated residual model; it is not a model-free probability. It assumes the threshold is known, the BQL status is correctly recorded, and the conditional observation distribution and its parameters are appropriate. In an NLME model, \(\mu\) generally depends on the population parameters and the subject’s random effects. The contribution is evaluated conditional on those random effects, and the subject likelihood then accounts for their distribution.

If a subject has several pre-quantification BQL records, and residual errors are conditionally independent given that subject’s model parameters, their conditional likelihood contributions multiply. Shared subject-level random effects still matter: after integrating over them, the records need not be independent in the marginal subject likelihood.

Why a BQL Record Constrains Early Exposure

At a fixed LLOQ and residual standard deviation, a larger predicted concentration makes a BQL outcome less probable. This can be seen directly from the likelihood contribution. Write \(z=(L-\mu)/\sigma\), and let \(\phi\) be the standard normal density. The derivative of the log BQL contribution with respect to \(\mu\) is

The inequality holds because \(\phi(z)>0\), \(\Phi(z)>0\), and \(\sigma>0\). Thus, conditional on the other model quantities, increasing the predicted concentration reduces the likelihood of that BQL record. This is the direct statistical constraint. Translating it into statements about absorption requires the concentration-time model.

For example, a one-compartment oral model with first-order absorption and elimination may be written, after a dose at time zero and for \(t>0\), as

Here \(D\) is dose, \(F\) is bioavailability, \(V\) is apparent volume, \(k_a\) is the absorption rate constant, and \(k_e\) is the elimination rate constant. This model starts at \(C(0)=0\) and, under the usual positive parameter conditions, can rise before declining. A pre-quantification BQL record therefore constrains the model’s predicted concentration at its particular early time. It does not, by itself, establish that absorption has not started. Nor does it isolate \(k_a\): changes in other parameters, including \(F/V\), can also change predicted concentrations.

Several Early BQL Records and a Later Quantified Sample

A sequence of pre-quantification BQL records can be more informative than one record, particularly if the model predicts concentrations above the LLOQ at several of those times. The likelihood combines the time-specific probabilities; it does not replace each BQL result with a guessed concentration. Later quantified values also matter because they anchor the subsequent profile and help distinguish a plausible delayed rise from a profile that is simply too low throughout.

Interpretation, Diagnostics, and Alternatives

In a population analysis, the likelihood contribution for an early BQL record depends on the individual predicted concentration, which in turn depends on population parameters, covariates, and subject-specific random effects. Across subjects, a systematic pattern of early BQL records that are assigned very small probabilities can pull estimates toward profiles with lower early predictions. This can be consistent with slower absorption or a delayed rise, but the BQL records alone do not prove either explanation. Parameter combinations can produce similar early predictions, and later quantified observations help determine which combinations remain plausible.

Useful checks focus on whether the fitted model reproduces the observed timing and frequency of early BQL records, and whether individual predictions make those statuses plausible. Review the actual sample times, dose time, LLOQ applied to each record, and BQL flags. Check that the first quantified sample is correctly identified for each subject. Examine whether BQL records occur systematically at times when the model predicts concentrations well above the limit, or whether the pattern is limited to a few subjects. A surprising early BQL record is not, on its own, proof that the whole model is wrong; repeated conflicts are more concerning.

The censored likelihood approach retains the information “below \(L\)” without inventing a concentration. Deleting BQL records discards that constraint; substituting a fixed value such as half the LLOQ or the LLOQ itself treats an unobserved value as though it had been measured. Those substitutions can alter the apparent early profile, while the censored contribution uses the probability implied by the observation model. The approach still depends on that model being adequate, and a small probability can be sensitive to the assumed residual variability.

A BQL record gives an inequality-type observation, not a precise concentration. If many early samples are BQL and only a few later values are quantified, the data may support a broad range of early profiles rather than a precise absorption estimate. Do not interpret the first quantified time as the time absorption began: quantification depends on the assay limit, the sampling schedule, and residual variability as well as on the underlying concentration curve.