Identifiability When Most Samples Are BQL

Why Extensive Censoring Can Leave Parameters Uncertain

A BQL record is not a measured concentration at zero or at the assay limit. It tells us that the underlying assay result fell below a threshold. When most samples are BQL, the data may therefore give a clear indication that concentrations are low without indicating how low. That distinction can matter greatly for disposition parameters such as clearance and volume.

The issue is not simply the proportion of BQL records. Information also depends on when samples were taken, the assay’s lower limit of quantification (LLOQ), residual variability, and whether the quantified concentrations distinguish the parameters of interest. A dataset with many late BQL records may still estimate an early exposure parameter well, while providing little information about the terminal decline.

A PK Model and a Censored-Data Likelihood

Consider a one-compartment model after an intravenous bolus dose \(D\). For individual \(i\), let \(CL_i\) be clearance and \(V_i\) be volume of distribution. The predicted concentration at time \(t\) is

A basic NLME parameterization allows individual disposition parameters to vary around population values: \(CL_i=CL_{\text{pop}}\exp(\eta_{CL,i})\) and \(V_i=V_{\text{pop}}\exp(\eta_{V,i})\), with the random effects modeled jointly, commonly as a mean-zero normal vector with covariance matrix \(\Omega\). For the calculations below, we condition on an individual’s parameters; estimating population parameters also requires accounting for the random-effects distribution.

Suppose the observation model has additive normal residual error, \(Y_{ij}=f_i(t_{ij})+\epsilon_{ij}\), where \(\epsilon_{ij}\sim N(0,\sigma^2)\). Assume the residual errors are independent conditional on the model parameters, the LLOQ is known, and a BQL record means precisely \(Y_{ij}<L\), where \(L\) is the LLOQ. A quantified value contributes its normal density; a BQL observation contributes a probability:

Here \(\Phi\) is the standard normal cumulative distribution function. This probability can be informative when the predicted concentration is near the LLOQ. If the prediction is far below the LLOQ, however, the BQL probability is close to one for many different predictions, so that record may contribute little information about the parameters.

Why the Information Can Collapse

The change in a BQL probability as a parameter changes makes the information loss precise. Let \(\theta\) be a scalar parameter, let \(f=f(\theta)\), and write \(p(\theta)=P(Y<L)=\Phi(z)\), where \(z=(L-f(\theta))/\sigma\). Differentiating gives

Here \(\phi\) is the standard normal density, and \(I(\theta)\) is the Bernoulli Fisher information in the censored indicator for this one observation. The expression follows because the indicator is Bernoulli with success probability \(p\): its score is \((B-p)(dp/d\theta)/[p(1-p)]\), whose expected squared value is the displayed information. When \(f\) lies far below \(L\), \(p\) approaches one and the probability changes very little for reasonable changes in \(f\); the information tends to zero. This explains why many deep-terminal BQL records can add little information about the rate of decline.

The argument applies to a single observation conditional on the parameters. With multiple conditionally independent observations, their log-likelihood contributions add. In an NLME model, population likelihood contributions also account for the distribution of individual random effects, rather than treating estimated individual effects as fixed known values.

What the Simulated Dataset Can and Cannot Identify

Consider a hypothetical simulated dataset with 12 subjects, each sampled at 6, 15, 21, 27, 33, and 39 hours after a 100 mg IV bolus. Suppose 12 of the 72 records are quantified at 6 hours and the remaining 60 are BQL. For a representative subject with \(V=20\) L and \(CL=8\) L/h, the predicted 6-hour concentration is \(5\exp(-2.4)=0.4536\) mg/L, while the 15-hour prediction is \(5\exp(-6)=0.01239\) mg/L. With an LLOQ of 0.050 mg/L, the later predictions are below the assay limit. This pattern can plausibly arise under the stated model, but it leaves little direct information about the exact concentrations in the late phase.

A useful geometric check is to first ignore residual error and consider only whether the model prediction is below the LLOQ. At 15 hours, the condition is

This is an approximate threshold illustration, not the actual normal-error likelihood: with residual error, BQL probability changes smoothly rather than switching at a sharp boundary. Still, it shows the basic limitation. A BQL result can rule against some predictions, but once predictions are sufficiently low, it does not say whether the concentration was slightly below the LLOQ or orders of magnitude below it.

There is also a parameter-separation problem. The curve depends on the amplitude \(D/V\) and the elimination rate \(CL/V\). One quantified concentration at one time supplies only one concentration constraint on these two quantities. For example, the 6-hour prediction of 0.4536 mg/L is produced by \(V=20\) L and \(CL=8\) L/h. It is also produced at that time by \(V=25\) L and \(CL\approx9.070\) L/h, since \(4\exp[-(9.070/25)\times6]\approx0.4536\). These parameter pairs predict different curves elsewhere, but when most later records only establish that the concentration is below the LLOQ, those records may not distinguish the curves sufficiently. In a population model, the same concern can extend to population parameters and random-effects variability.

Weak Identification or Optimizer Failure?

An optimizer can stop at a poor solution, but a successfully converged optimizer cannot create information that the data do not contain. The distinction matters: changing algorithms may resolve a numerical failure, while it cannot make a broad likelihood profile narrow. Evaluate both the numerical behavior and the shape of the likelihood.

These signs are not individually decisive. Flat profiles can make standard errors unstable, and numerical problems can occur in a weakly identified model. Report the evidence: convergence behavior, sensitivity to starting values, profile shape, and the resulting uncertainty. Do not label every imprecise estimate an optimizer failure, or every converged estimate well identified.

Design Implications, Alternatives, and Limits

For disposition parameters, the most useful response to weak information is often to improve the information content of sampling and measurement. Sampling at times that capture both the concentration scale and the decline can help distinguish \(D/V\) from \(CL/V\); choosing an assay with a lower LLOQ can make later concentrations quantifiable. These are design considerations, not guarantees: parameter separation still depends on the model and the full pattern of observations.

A likelihood that uses quantified values and BQL probabilities retains more of the recorded information than replacing BQL values with an invented concentration. Earlier material on M1–M7 distinguishes approaches to BQL records; for identifiability, the essential point is that the assumed likelihood must correspond to what the record actually says. Substitution changes the data supplied to the model and can alter apparent parameter precision as well as estimates. Conversely, a correctly specified censored likelihood cannot recover an unobserved concentration below the LLOQ or ensure precise estimates when most records are uninformative.

Practical identification depends on assumptions. Misspecified residual error, an incorrect structural model, inaccurate assay limits, or dependence not represented in the likelihood can make profiles misleading. The simple calculations here use additive normal error and condition on individual parameters; a lognormal observation model or integrated NLME likelihood changes the calculation, although the general principle remains: when the probability of a BQL result is nearly unchanged across parameter values, that observation offers little discrimination. The analysis also concerns parameter information, not whether a particular method of handling BQL data produces bias.