Handling BLQ PK Data Across Staged and Joint Fits

Why a BLQ PK record still contains information

A concentration reported as below the limit of quantification (BLQ) is not an observed concentration equal to zero, nor is it an exact value at the assay limit. It tells us that the assay result fell in a known region: below the lower limit of quantification. The record therefore constrains plausible PK predictions, even though it does not specify a precise concentration. How that constraint is carried into the PD fit can affect estimated exposure and, in turn, estimated drug effect.

The key distinction is between replacing the censored value with a number and using the probability of observing a BLQ result under the model. The latter keeps the observation at its actual level of precision. In a staged analysis, the PK fit is completed first and its results are then passed to the PD fit. A joint analysis estimates PK and PD parameters together, using both types of observations in one likelihood. Neither label alone guarantees a sound analysis: the observation model, handling of censoring and transfer of uncertainty all matter.

Likelihood contribution under a lognormal observation model

Suppose concentrations are positive and the residual error is additive on the log scale:

Here \(C\) is the conditional median of \(Y\), and \(\sigma\) is the residual standard deviation on the log scale. The model assumes the stated lognormal distribution is appropriate, the limit \(L\) is known for the record, and the BLQ classification accurately means the measured value is below that limit. Then a BLQ record has likelihood contribution

where \(\Phi\) is the standard normal cumulative distribution function. For a quantified value \(y\geq L\), the contribution is instead the lognormal density at \(y\). In a population model, an individual’s concentration may depend on parameters \(\theta\) and random effects \(\eta_i\). A conditional likelihood uses the probability given \(\eta_i\); a marginal likelihood integrates that conditional contribution over the random-effects distribution. Whichever fitting framework is used, the BLQ contribution is a probability, not a density evaluated at an invented concentration.

This formula follows by standardizing the log concentration. Since the logarithm is increasing, \(Y<L\) is equivalent to \(\log Y<\log L\). Subtracting the conditional mean and dividing by \(\sigma\) gives

A different residual model requires its own corresponding probability. For example, with additive normal error on the original concentration scale, the probability is calculated from that model’s normal distribution, not from the lognormal formula above. The censoring likelihood is only as credible as the observation model and the recording process it represents.

Staged handling: substitution versus censored PK fitting

A staged analysis first fits the PK model and then uses the fitted PK results in the PD analysis. One shortcut is to substitute a fixed value, such as zero, \(L/2\), or \(L/\sqrt{2}\), for each BLQ result. The PK fit then treats that value as if it were measured exactly. This changes the data: it turns a range statement into an artificial point and usually understates the uncertainty about the latent concentration. The direction and size of parameter bias depend on the model, timing of BLQ records, censoring frequency and chosen substitute; no single correction factor applies in general.

A more principled staged approach fits the PK model with the censored likelihood contribution \(P(Y<L\mid C)\). This retains the distinction between exact quantified values and threshold-only observations. However, a staged PD fit may still condition on point estimates of the PK parameters or individual exposures. If uncertainty in those estimates is ignored, PD uncertainty can be understated; and if the PD relationship is nonlinear in exposure, plugging in a point estimate can also shift the PD estimate. These are consequences of the staging and information transfer, not errors in the BLQ probability itself.

The conditional mean of concentration among BLQ outcomes also illustrates why the limit itself is not an imputed value. Under the lognormal model above,

This is a model-based conditional mean, not a universal replacement rule. It depends on \(C\), \(\sigma\), and \(L\), and using it as a fixed value would still discard the remaining uncertainty and the fact that the observation entered the data only through a censoring event.

Joint likelihood and a comparison of approaches

In a joint fit, the BLQ PK contribution and the PD contribution enter the likelihood together. For example, if \(R\) is a PD observation with conditional density \(f_R\), a simple conditional contribution for one BLQ PK record and one PD record is

where \(\theta\) denotes PK parameters, \(\psi\) denotes PD parameters, and \(\eta_i\) denotes the relevant individual effects. This product assumes the PK and PD measurement errors are conditionally independent given the model quantities shown. If that assumption is not appropriate, their joint observation model must represent the dependence. In a population likelihood, random effects are integrated over their distribution as specified by the model.

The joint fit can allow PD observations to inform PK parameters when PK data alone are weakly informative and the PK/PD link is correctly specified. It does not make a BLQ concentration exact, nor does it remove the need to model the censoring event. It can also transmit PK misspecification into PD estimates, or vice versa. A staged censored PK fit followed by a PD fit is easier to separate and inspect, but can lose information or under-represent uncertainty if only point estimates are transferred. Substitution is computationally simple, but relies on an arbitrary value and is generally the least faithful representation of a threshold observation.

Diagnostics, failure modes and limits

Check first that each record’s assay limit and BLQ status are correctly represented. Limits can differ across assays, batches or samples; applying one threshold indiscriminately changes the censoring probabilities. Check that the residual model is plausible near the limit, where a large fraction of the information may come from censored observations. A lognormal model, for instance, cannot accommodate a true point mass at zero without an additional model component.

Useful diagnostics include comparing observed BLQ frequencies across time or dose groups with frequencies predicted by the fitted model, examining quantified residuals separately from censored records, and checking sensitivity to reasonable observation-model choices. A model that predicts almost no BLQ records in a group where many occur, or the reverse, may have a misspecified PK trajectory, residual error, assay handling or population variability. A high fraction of BLQ values can leave PK parameters weakly identified; a likelihood contribution can be valid yet provide little information to distinguish nearby parameter values.

For staged fits, compare PD estimates and uncertainty under a censored PK fit with those from any substitution-based analysis, and document the substitution rule if it is retained as a sensitivity analysis. For joint fits, inspect whether PD data are exerting unexpectedly strong influence on PK estimates, especially when the PK/PD relationship is poorly supported by observed concentrations. These comparisons are diagnostics, not automatic rules for selecting a method. Joint fitting cannot repair a wrong structural model, an inappropriate residual distribution, or an incorrect conditional-independence assumption. Conversely, a staged analysis is not inherently invalid if it preserves the censoring likelihood and handles uncertainty transfer adequately for the intended inference.